On the classification of primitive ideals for complex
classical Lie algebras, IV

DEVRA GARFINKLE

THOMAS PIETRAHO
Department of Mathematics, Bowdoin College, Brunswick, ME 04011, USA

W. M. MCGOVERN
Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

## Introduction

This paper is the fourth and last in the series “On the classification of
primitive ideals for complex classical Lie algebras", extending the main
results of [5] to type $D$. Here the generalized $\tau$-invariant defined in
[5] must be defined in a different way, as there is no pair of adjacent simple
roots in type $D$ with different lengths. Instead we attach a family of
operators to a quadruple of simple roots spanning a subsystem of type $D_{4}$,
each of them taking either one or two values, like the corresponding operators
in types $B$ and $C$, and use these to characterize primitive ideals via their
generalized $\tau$-invariants.

##

###### 4.1.1. NOTATION.

We will implicitly rely on the notation introduced in the first three parts of
this sequence of papers. It is summarized in an appendix of [5].

  1. (1)

For simple roots we take $\alpha_{1}^{\prime}=e_{2}+e_{1}$ and
$\alpha_{m}=e_{m}-e_{m-1}$ for $2\leq m\leq n$. This non-standard notation
will facilitate the use of results from [5]. Set
$\Pi^{\prime}=\\{\alpha_{1}^{\prime},\alpha_{2},\ldots,\alpha_{n}\\}$ and
write $s_{1}^{\prime}$ for $s_{\alpha_{1}^{\prime}}$, the reflection in
$\alpha_{1}^{\prime}$. We let $W$ be the Weyl group of type $C_{n}$ and will
write $W^{\prime}$ for the subgroup of $W$ generated by the reflections in the
roots belonging to $\Pi^{\prime}$. Further, let $W^{\prime\prime}=W\setminus
W^{\prime}$.

  2. (2)

We define

$\mathcal{S}^{\prime}(M_{1},M_{2})=\\{\gamma\in\mathcal{S}(M_{1},M_{2})\,|\,|\\{(e,g,\epsilon)\in\gamma\,|\epsilon=-1\\}|\equiv
0\pmod{2}\\},$

and set
$\mathcal{S}^{\prime\prime}(M_{1},M_{2})=\mathcal{S}(M_{1},M_{2})\setminus\mathcal{S}^{\prime}(M_{1},M_{2})$.
Define $\mathcal{S}^{\prime}(n,n)$ and $\mathcal{S}^{\prime\prime}(n,n)$ by
imitating [3, 1.1.2].

  3. (3)

For $(\mathbf{T}_{1},\mathbf{T}_{2})\in\mathcal{T}(M_{1},M_{2})$ we write
$n_{h}((\mathbf{T}_{1},\mathbf{T}_{2}))=n_{h}(\mathbf{T}_{1})+n_{h}(\mathbf{T}_{2})$,
and similarly define $n_{v}((\mathbf{T}_{1},\mathbf{T}_{2}))$.

  4. (4)

We define

$\mathcal{T}^{\prime}(M_{1},M_{2})=\\{(\mathbf{T}_{1},\mathbf{T}_{2})\in\mathcal{T}(M_{1},M_{2})\,|\,n_{v}((\mathbf{T}_{1},\mathbf{T}_{2}))\equiv
0\pmod{4}\\}$

and

$\mathcal{T}^{\prime\prime}(M_{1},M_{2})=\\{(\mathbf{T}_{1},\mathbf{T}_{2})\in\mathcal{T}(M_{1},M_{2})\,|\,n_{v}((\mathbf{T}_{1},\mathbf{T}_{2}))\equiv
2\pmod{4}\\}.$

The sets $\mathcal{T^{\prime}}(n,n)$ and $\mathcal{T}^{\prime\prime}(n,n)$ are
defined by imitating [3, 1.1.9]. When we wish to specify that the $D$ grid is
being used explicitly, we will write $\mathcal{T}^{\prime}_{D}(M_{1},M_{2})$,
$\mathcal{T}^{\prime\prime}_{D}(M_{1},M_{2})$,
$\mathcal{T}^{\prime}_{D}(n,n),$ and $\mathcal{T}^{\prime\prime}_{D}(n,n)$.
Write $\mathcal{T}_{D}(n,n)$ for the union of $\mathcal{T}^{\prime}_{D}(n,n)$
and $\mathcal{T}^{\prime\prime}_{D}(n,n)$.

###### 4.1.2. REMARK.

For $w\in W$ we have
$\delta(s_{1}^{\prime}w)=\textup{SC}^{R}(1;\textup{SC}^{R}(2;\textup{In}^{R}(1,2;\delta(w))))$
and
$\delta(ws_{1}^{\prime})=\textup{SC}^{L}(1;\textup{SC}^{L}(2;\textup{In}^{L}(1,2;\delta(w))))$.
Consequently, we have

$\delta(W^{\prime})=\mathcal{S}^{\prime}(n,n)\text{ and
}\delta(W^{\prime\prime})=\mathcal{S}^{\prime\prime}(n,n).$

###### 4.1.3. PROPOSITION.

Let $(\mathbf{T}^{\prime},v,\epsilon)\in M$ and let
$(\mathbf{T},P)=\alpha((\mathbf{T}^{\prime},v,\epsilon))$.

  1. (1)

If $\epsilon=1$, then either

    1. (a)

$P$ is horizontal, $n_{h}(\mathbf{T})=n_{h}(\mathbf{T}^{\prime})+1$, and
$n_{v}(\mathbf{T})=n_{v}(\mathbf{T}^{\prime})$, or

    2. (b)

$P$ is vertical, $n_{h}(\mathbf{T})=n_{h}(\mathbf{T}^{\prime})+2$, and
$n_{v}(\mathbf{T})=n_{v}(\mathbf{T}^{\prime})-1$.

  2. (2)

If $\epsilon=-1$, then either

    1. (a)

$P$ is horizontal, $n_{h}(\mathbf{T})=n_{h}(\mathbf{T}^{\prime})-1$, and
$n_{v}(\mathbf{T})=n_{v}(\mathbf{T}^{\prime})+1$, or

    2. (b)

$P$ is vertical, $n_{h}(\mathbf{T})=n_{h}(\mathbf{T}^{\prime})$, and
$n_{v}(\mathbf{T})=n_{v}(\mathbf{T}^{\prime})+2$.

###### Proof.

The proof is by induction on $|M|$; the case when $|M|=1$ is obvious. Set
$e=\sup M$ and note that the proof of the proposition is clear when $v=e$, so
assume $v\neq e$. Set $\overline{\mathbf{T}}^{\prime}=\mathbf{T}^{\prime}-e$
and
$(\overline{\mathbf{T}},\overline{P})=\alpha((\overline{\mathbf{T}}^{\prime},v,\epsilon)).$
Then $\overline{\mathbf{T}}=\mathbf{T}-e.$ Set $P_{e}=P(e,\mathbf{T})$ and
$P_{e}^{\prime}=P(e,\mathbf{T}^{\prime})$. We will prove statement (1) for
$n_{h}$, the others can be verified by a similar case-by-case analysis. There
are four cases.

  1. Case A.

Here $\overline{P}$ is horizontal and $|\overline{P}\cap{P}_{e}^{\prime}|=1$.
Then $P_{e}^{\prime}$ and $P$ are vertical and $P_{e}$ is horizontal. Thus
$n_{h}(\mathbf{T})=n_{h}(\overline{\mathbf{T}})+1=n_{h}(\overline{\mathbf{T}}^{\prime})+2=n_{h}(\mathbf{T}^{\prime})+2$,
as desired.

  2. Case B.

Here $\overline{P}$ is horizontal and
$\overline{P}=\overline{P}_{e}^{\prime}$. Then $P_{e}^{\prime}$, $P$, and
$P_{e}$ are horizontal. Thus
$n_{h}(\mathbf{T})=n_{h}(\overline{\mathbf{T}})+1=n_{h}(\overline{\mathbf{T}}^{\prime})+2=n_{h}(\mathbf{T}^{\prime})+2$.

  3. Case C.

Here $\overline{P}$ is vertical and $|\overline{P}\cap{P}_{e}^{\prime}|=1$.
Then $P_{e}^{\prime}$ and $P$ are horizontal and $P_{e}$ is vertical. Thus
$n_{h}(\mathbf{T})=n_{h}(\overline{\mathbf{T}})=n_{h}(\overline{\mathbf{T}}^{\prime})+2=n_{h}(\mathbf{T}^{\prime})+1$.

  4. Case D.

Here $\overline{P}$ is vertical and $\overline{P}=\overline{P}_{e}^{\prime}$.
Then $P_{e}^{\prime}$, $P$, and $P_{e}$ are vertical. Thus
$n_{h}(\mathbf{T})=n_{h}(\overline{\mathbf{T}})=n_{h}(\overline{\mathbf{T}}^{\prime})+2=n_{h}(\mathbf{T}^{\prime})+2$.

∎

###### 4.1.4. COROLLARY.

Let $\gamma\in\mathcal{S}(M_{1},M_{2})$. Then $n_{h}(A(\gamma))\equiv
0\pmod{2}$ and $n_{v}(A(\gamma))\equiv 0\pmod{2}.$ We have
$\gamma\in\mathcal{S}^{\prime}(M_{1},M_{2})$ if and only if
$n_{v}(A(\gamma))\equiv 0\pmod{4}$ and
$\gamma\in\mathcal{S}^{\prime\prime}(M_{1},M_{2})$ if and only if
$n_{v}(A(\gamma))\equiv 2\pmod{4}.$

###### Proof.

The proof is by induction on $|M_{1}|$, and is then obvious from the
definitions and Proposition 4.1.3. ∎

###### 4.1.5. COROLLARY.

  1. (1)

We have
$\mathcal{T}(M_{1},M_{2})=\mathcal{T}^{\prime}(M_{1},M_{2})\cup\mathcal{T}^{\prime\prime}(M_{1},M_{2})$.

  2. (2)

The map $A$ yields a bijection from $\mathcal{S}^{\prime}(M_{1},M_{2})$ to
$\mathcal{T}^{\prime}(M_{1},M_{2})$ and from
$\mathcal{S}^{\prime\prime}(M_{1},M_{2})$ to
$\mathcal{T}^{\prime\prime}(M_{1},M_{2})$.

###### 4.1.6. COROLLARY.

The map $A\circ\delta$ yields a bijection from $W^{\prime}$ to
$\mathcal{T}^{\prime}(n,n)$ and from $W^{\prime\prime}$ to
$\mathcal{T}^{\prime\prime}(n,n)$.

##

###### 4.2.1. NOTATION.

Let $(\mathbf{T}_{1},\mathbf{T}_{2})\in\mathcal{T}_{K}(M_{1},M_{2})$ for
$K=B,$ $C$, or $D$. Let $c$ be an extended cycle in $\mathbf{T}_{1}$ relative
to $\mathbf{T}_{2}$ and suppose that $c$ is a union of open cycles in
$\mathbf{T}_{1}$. We write

$Cy(c)=\\{c^{\prime}\;|\;c^{\prime}\subseteq c\text{ and
}c^{\prime}\in\textup{OC}(\mathbf{T}_{1})\\},$

and

$Sq(c)=\\{S\;|\;\text{there is a }c^{\prime}\in Cy(c)\text{ such that
}S=S_{b}(c^{\prime})\text{ or }S=S_{f}(c^{\prime})\\}.$

We let $n_{u}(c)$ be the number of up cycles contained in $c$ and $n_{d}(c)$
be the number of down cycles contained in $c$.

###### 4.2.2. PROPOSITION.

Let $(\mathbf{T}_{1},\mathbf{T}_{2})$ and $c$ be as in 4.2.1 and suppose that
$Cy(c)\subseteq\textup{OC}^{*}(\mathbf{T}_{1})$. Let $\overline{c}$ be the
extended cycle in $\mathbf{T}_{2}$ relative to $\mathbf{T}_{1}$ corresponding
to $c$. Then either
$n_{u}(c)+n_{u}(\overline{c})=n_{d}(c)+n_{d}(\overline{c})+2$ or
$n_{u}(c)+n_{u}(\overline{c})=n_{d}(c)+n_{d}(\overline{c})-2$.

###### Proof.

The proof is by induction on $|M_{1}|$, the proposition being vacuously true
when $|M_{1}|=0.$ Let $e=\sup M_{1}$. If $e\notin c$ then the proposition in
true by induction using Theorem 2.2.3 and Proposition 2.2.4, cf. Proposition
2.3.3(b)). So assume $e\in c$. The proposition is clear if $c=\\{e\\}$ so
assume $c\neq\\{e\\}$. Let
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})=(\mathbf{T}_{1},\mathbf{T}_{2})-L$.
Then by Proposition 2.3.3(b), $c\setminus\\{e\\}$ is an extended cycle in
$\mathbf{T}_{1}^{\prime}$ relative to $\mathbf{T}_{2}^{\prime}$. Set
$c^{\prime}=c\setminus\\{e\\}$ and let $\overline{c}^{\prime}$ be the
corresponding extended cycle in $\mathbf{T}_{2}^{\prime}$ relative to
$\mathbf{T}_{1}^{\prime}$. We will assume that $P(e,\mathbf{T}_{1})$ is
horizontal; an analogous transposed argument works when $P(e,\mathbf{T}_{1})$
is vertical. If $P^{\prime}(e,T_{1})$ is also horizontal, that is, if for some
$\tilde{c}\in Cy(c)$, $\tilde{c}\neq\\{e\\}$, we have $S_{b}(\tilde{c})\in
P(e,\mathbf{T}_{1})$ or $S_{f}(\tilde{c})\in P^{\prime}(e,\mathbf{T}_{1})$,
then the proposition is clear using the c.s.p.b’s of Theorem 2.2.3-(1a) or
(2a) and Proposition 2.2.4-(1a) or (2a). So assume
$P^{\prime}(e,\mathbf{T}_{1})$ is vertical. Set
$P(e,\mathbf{T}_{1})=\\{S_{ij},S_{i,j+1}\\}.$ There are two cases:

  1. Case A.

Here $\\{e\\}\in Cy(c)$. Then
$P^{\prime}(e,\mathbf{T}_{1})=\\{S_{ij},S_{i+1,j}\\}.$ Thus $\\{e\\}$ is a
down cycle, and so by using Proposition 2.2.4 (1b) we have
$n_{u}(c)=n_{u}(c^{\prime})$ and $n_{d}(c)=n_{d}(c^{\prime})+1.$ Now by
Theorem 2.2.3 (1b) there are cycles $c^{1}$ and
$c^{2}\in\textup{OC}^{*}(\mathbf{T}_{2})$ and
$\tilde{c}^{\prime}\in\textup{OC}^{*}(\mathbf{T}_{2}^{\prime})$ such that
$S_{b}(c^{1})=S_{b}(\tilde{c}^{\prime})$, $S_{f}(c^{1})=S_{i+1,j}$,
$S_{b}(c^{2})=S_{i,j+1}$, and $S_{f}(c^{2})=S_{f}(\tilde{c}^{\prime}).$ Note
that the alternative suggested by Theorem 2.2.3 (1b) is the existence of a
cycle $\tilde{c}\in\textup{OC}^{*}(\mathbf{T}_{2})$ with
$S_{b}(\tilde{c})=S_{i+1,j}$ and $S_{f}(\tilde{c})=S_{i,j+1}$, but then
$\\{e\\}$ is an extended cycle in $\mathbf{T}_{1}$ relative to
$\mathbf{T}_{2}$, contradicting the assumption that $c\neq\\{e\\}$. Let
$U=\mu(Cy(\overline{c})\setminus\\{c^{1},c^{2}\\})$ where $\mu$ is the
c.s.p.b. of Theorem 2.2.3 (1b). Then
$\overline{c}^{\prime}=(\bigcup_{\tilde{c}\in
U}\tilde{c})\cup\tilde{c}^{\prime}.$ There are three possibilities. Suppose
first that $c^{1}$ is a down cycle. Then $c^{2}$ is nested in $c^{1}$ and thus
we have
$\rho(S_{b}(\tilde{c}^{\prime}))=\rho(S_{b}(c^{1}))<\rho(S_{f}(c^{2}))<i$, so
$c^{2}$ is an up cycle and $\tilde{c}^{\prime}$ is a down cycle. Similarly, if
$c^{2}$ is a down cycle, then $c^{1}$ is an up cycle and $\tilde{c}^{\prime}$
is a down cycle. Finally, if both $c^{1}$ and $c^{2}$ are up cycles then
$\tilde{c}^{\prime}$ is an up cycle. In all three cases we have
$n_{u}(\overline{c})=n_{u}(\overline{c}^{\prime})+1$ and
$n_{d}(\overline{c})=n_{d}(\overline{c}^{\prime})$. Thus

$n_{u}(c)+n_{u}(\overline{c})=n_{u}(c^{\prime})+n_{u}(\overline{c}^{\prime})+1\text{
and
}n_{d}(c)+n_{d}(\overline{c})=n_{d}(c^{\prime})+n_{d}(\overline{c}^{\prime})+1,$

so the proposition holds by induction.

  2. Case B.

Here $\\{e\\}$ is not a cycle in $\mathbf{T}_{1}$. Then
$P^{\prime}(e,\mathbf{T}_{1})=\\{S_{i-1,j+1},S_{i,j+1}\\}$. By Theorem
2.2.3-(2b), there is a cycle
$\tilde{c}^{\prime}\in\textup{OC}^{*}(\mathbf{T}_{2}^{\prime})$ such that
$S_{b}(\tilde{c}^{\prime})=S_{i-1,j+1}$ and
$S_{f}(\tilde{c}^{\prime})=S_{ij}$. In particular $\tilde{c}^{\prime}$ is a
down cycle so the c.s.p.b. of Theorem 2.2.3-(2b) implies
$n_{u}(\overline{c})=n_{u}(\overline{c}^{\prime})$ and
$n_{d}(\overline{c})=n_{d}(\overline{c}^{\prime})-1$. By Proposition
2.2.4-(2b), there are cycles $\tilde{c}\in\textup{OC}^{*}(\mathbf{T}_{1})$ and
$c^{1}$, $c^{2}\in\textup{OC}^{*}(\mathbf{T}_{1}^{\prime})$ such that
$S_{b}(c^{1})=S_{b}(\tilde{c})$, $S_{f}(c^{1})=S_{ij}$,
$S_{b}(c^{2})=S_{i-1,j+1}$, and $S_{f}(c^{2})=S_{f}(\tilde{c})$. We now
proceed as in the previous case, finding $n_{u}(c)=n_{u}(c^{\prime})-1$ and
$n_{d}(c)=n_{d}(c^{\prime})$. Thus again the proposition is true by induction.

∎

###### 4.2.3. DEFINITION.

Let $(\mathbf{T}_{1},\mathbf{T}_{2})\in\mathcal{T}_{K}(M_{1},M_{2})$ for
$K=B$, $C$, or $D$. Let $c$ be an extended cycle in $\mathbf{T}_{1}$ relative
to $\mathbf{T}_{2}$ and suppose either $c$ is an closed cycle in
$\mathbf{T}_{1}$ or $Cy(c)\subseteq\textup{OC}^{*}(\mathbf{T}_{1})$. We say
that $c$ is an up extended cycle is either $c$ is a closed up cycle in
$\mathbf{T}_{1}$ or
$n_{u}(c)+n_{u}(\overline{c})=n_{d}(c)+n_{d}(\overline{c})+2$. Otherwise, we
say that $c$ is a down extended cycle.

###### 4.2.4. PROPOSITION.

Let $(\mathbf{T}_{1},\mathbf{T}_{2})$ and $c$ be as in Definition 4.2.3. Set
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})=\mathbf{E}((\mathbf{T}_{1},\mathbf{T}_{2}),c,L).$

  1. (1)

If $c$ is an up extended cycle, then

$n_{h}((\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime}))=n_{h}((\mathbf{T}_{1},\mathbf{T}_{2}))+2\text{
and
}n_{v}((\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime}))=n_{v}((\mathbf{T}_{1},\mathbf{T}_{2}))-2.$

  2. (2)

If $c$ is a down extended cycle, then

$n_{v}((\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime}))=n_{v}((\mathbf{T}_{1},\mathbf{T}_{2}))+2\text{
and
}n_{h}((\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime}))=n_{h}((\mathbf{T}_{1},\mathbf{T}_{2}))-2.$

###### Proof.

If $c$ is a closed cycle in $\mathbf{T}_{1}$, then this is given by conditions
(1) and (2) of Propositions 3.3.10 and 3.3.11. If $c$ is a union of open
cycles in $\mathbf{T}_{1}$ with
$Cy(c)\subseteq\textup{OC}^{*}(\mathbf{T}_{1})$, then this proposition is a
consequence of Proposition 4.2.2 and conditions (3) and (4) of Propositions
3.3.7 and 3.3.8. ∎

###### 4.2.5. REMARK.

We have now proved the appropriate analogues of conditions (3) and (4) of
Propositions 3.3.7 and 3.3.8 for up and down extended cycles $c$ with
$Cy(c)\subseteq\textup{OC}^{*}(\mathbf{T}_{1})$. The relevant condition (5) of
both propositions is also true via induction. Conditions (1) and (2) can be
modified as below. We state the result for up extended cycles; analogous
statements hold for down extended cycles.

###### 4.2.6. PROPOSITION.

Let $(\mathbf{T}_{1},\mathbf{T}_{2})\in\mathcal{T}_{K}(M_{1},M_{2})$ for
$K=B$, $C$, or $D$ and let $c$ be an up extended cycle in $\mathbf{T}_{1}$
relative to $\mathbf{T}_{2}$. If $\hat{c}\in Cy(c)$, then either $\hat{c}$ is
an up cycle or there is an up cycle $\hat{c}^{1}\in Cy(c)$ with $\hat{c}$
nested in $\hat{c}^{1}$. Let $k=|Cy(c)|$ and label the squares
$Sq(c)=\\{S_{1},\ldots,S_{2k}\\}$ so that $\rho(S_{1})<\ldots<\rho(S_{2k}),$
or equivalently, $\kappa(S_{1})>\ldots>\kappa(S_{2k})$. Then

  1. (1)

$S_{1}=S_{f}(c^{\prime})$ for some $c^{\prime}\in Cy(c)$ and
$S_{2k}=S_{b}(c^{\prime\prime})$ for some $c^{\prime\prime}\in Cy(c)$, and

  2. (2)

$S_{i}=S_{f}(c^{\prime})$ for some $c^{\prime}\in Cy(c)$ if and only if
$S_{i+1}=S_{b}(c^{\prime\prime})$ for some $c^{\prime\prime}\in Cy(c)$.

###### Proof.

The proof is by induction as for Proposition 4.2.2. The only difficulty arises
in Case A, so let $e$, $(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})$,
$c^{1}$, $c^{2},$ and $c^{\prime}$ be as therein. Assume first that $c^{1}$ is
an up cycle. Note that there is an even number of squares $S\in
Sq(c^{\prime})$ with $i+1<\rho(S)<\rho(S_{b}(c^{1}))$, since any cycle in
$\mathbf{T}_{2}$ with such an $S$ as its forward or back square is nested in
$c^{1}$. By (2) and induction, these alternate between back and forward
squares, so if we pick $S\in Sq(c^{\prime})$ with $\rho(S)$ maximal given that
$\rho(S)\leq\rho(S_{b}(c^{1}))$, then there is a $\hat{c}^{2}\in
Cy(c^{\prime})$ such that $S=S_{b}(\hat{c}^{2}).$ If $\hat{c}^{2}$ is an up
cycle, the $\hat{c}=\\{e\\}$ is nested in $\hat{c}^{2}$, otherwise by
induction $\hat{c}^{2}$ is nested in some up cycle in $c^{\prime}$ which then
also nests $\\{e\\}$. We see that statement (2) is also satisfied by $c$. The
proof is similar when $c^{2}$ is an up cycle. Finally, as in Proposition
4.2.2, we must also consider the proof of the transpose of Case A. This is
trivial for (1), and for (2), it is the transposed proof.

∎

##

###### 4.3.1. DEFINITION.

Let $w\in W^{\prime}$ or $w\in W^{\prime\prime}$. We define $\tau^{L}(w)$,
$\tau^{R}(w)\subseteq\Pi^{\prime}$ as in Definition 2.1.3 using
$\alpha^{\prime}$ and $s_{1}^{\prime}$ in place of $\alpha_{1}$ and $s_{1}$.
We define $D_{\alpha\beta}^{L}(W^{\prime})$ and
$D_{\alpha\beta}^{R}(W^{\prime})$ for any pair of adjacent simple roots
$\alpha,\beta\in\Pi^{\prime}$ as in Definition 2.1.4 (1) and
$T_{\alpha\beta}^{L}$ and $T_{\alpha\beta}^{R}$ as in Definition 2.1.4-(2).
Note that $\tau^{R}(w)$ coincides with the usual notion of the
$\tau$-invariant of $w$ [7, 2.6], while $T_{\alpha\beta}^{R}$ coincides with
the usual definition of $T_{\alpha\beta}$. Note also that the map $\delta$
taking an element $w$ of $W$ to a finite sequence of triples, from which the
tableaux $\mathbf{L}(w),\mathbf{R}(w)$ are constructed, interchanges
$\tau^{L}$ and $\tau^{R}$, by the remark after Definition 2.1.7.

###### 4.3.2. DEFINITION.

Let $\gamma\in\mathcal{S}(M_{1},M_{2})$.

  1. (1)

Suppose $\\{1,2\\}\subseteq M_{1}$. Let $k$, $l$, $\epsilon_{1}$, and
$\epsilon_{2}$, be such that
$\\{(1,k,\epsilon_{1}),(2,l,\epsilon_{2})\\}\subseteq\gamma.$ We say that
$\alpha_{1}^{\prime}\in\tau^{L}(\gamma)$ if either both $k>l$ and
$\epsilon_{1}=-1$ or both $k<l$ and $\epsilon_{2}=-1$; otherwise we say
$\alpha_{1}^{\prime}\notin\tau^{L}(\gamma)$. We define analogously
$\alpha_{1}^{\prime}\in\tau^{R}(\gamma)$ and
$\alpha_{1}^{\prime}\notin\tau^{R}(\gamma)$ when $\\{1,2\\}\subseteq M_{2}$.

  2. (2)

For $i\geq 2$, the statements $\alpha_{i}\in\tau^{L}(\gamma)$,
$\alpha_{i}\notin\tau^{L}(\gamma)$, $\alpha_{i}\in\tau^{R}(\gamma)$, and
$\alpha_{i}\notin\tau^{R}(\gamma)$ are defined as in Definition 2.1.5.

  3. (3)

Suppose that either $\\{1,2,3\\}\subseteq M_{1}$ and
$\\{\alpha,\beta\\}=\\{\alpha_{1}^{\prime},\alpha_{3}\\}$ or
$\\{i-1,i,i+1\\}\subseteq M_{1}$ and
$\\{\alpha,\beta\\}=\\{\alpha_{i},\alpha_{i+1}\\}$. Then we define
$D_{\alpha\beta}^{L}(\mathcal{S}(M_{1},M_{2}))$ and
$D_{\alpha\beta}^{R}(\mathcal{S}(M_{1},M_{2}))$ as in Definition 2.1.7 (1).

  4. (4)

If $\\{\alpha,\beta\\}=\\{\alpha_{1}^{\prime},\alpha_{3}\\}$ and
$\\{1,2,3\\}\subseteq M_{1}$ , we define

$T^{L}_{\alpha\beta}:D_{\alpha\beta}^{L}(\mathcal{S}(M_{1},M_{2}))\longrightarrow
D_{\beta\alpha}^{L}(\mathcal{S}(M_{1},M_{2}))$

by setting $T_{\alpha\beta}^{L}(\gamma)=\gamma^{\prime}$ where

$\\{\gamma^{\prime}\\}=\\{\textup{SC}^{L}(1;\textup{SC}^{L}(2;\textup{In}(1,2;\gamma))),\textup{In}^{L}(2,3;\gamma)\\}\cap
D_{\beta\alpha}^{L}(\mathcal{S}(M_{1},M_{2})).$

As usual, $T_{\alpha\beta}^{L}$ is a well-defined bijection with inverse
$T_{\beta\alpha}^{L}$.

  5. (5)

If $\\{i-1,i,i+1\\}\subseteq M_{1}$ and
$\\{\alpha,\beta\\}=\\{\alpha_{i},\alpha_{i+1}\\}$ with $i\geq 2$, we define

$T^{L}_{\alpha\beta}:D_{\alpha\beta}^{L}(\mathcal{S}(M_{1},M_{2}))\longrightarrow
D_{\beta\alpha}^{L}(\mathcal{S}(M_{1},M_{2}))$

as in Definition 2.1.7-(2).

  6. (6)

We define analogously the operators

$T^{R}_{\alpha\beta}:D_{\alpha\beta}^{R}(\mathcal{S}(M_{1},M_{2}))\longrightarrow
D_{\beta\alpha}^{R}(\mathcal{S}(M_{1},M_{2})).$

###### REMARK.

The following properties of the $\tau$-invariant also hold with $R$ and $L$
interchanged:

  1. (1)

We have $\tau^{R}(\delta(w))=\tau^{L}(w)$ and for $w\in
D_{\alpha\beta}^{L}(W^{\prime})$ we have
$\delta(T_{\alpha\beta}^{L}(w))=T_{\alpha\beta}^{R}(\delta(w))$.

  2. (2)

If $\gamma\in\mathcal{S}(M_{1},M_{2})$ and $\\{1,2\\}\subseteq M_{1}$, then
$\alpha_{1}^{\prime}\in\tau^{L}(\gamma)$ if and only if
$\alpha_{1}^{\prime}\notin\tau^{L}(\textup{SC}(\gamma))$.

  3. (3)

If $\gamma\in D_{\alpha\beta}^{L}(\mathcal{S}(M_{1},M_{2}))$, then
$\textup{SC}(\gamma)\in D_{\beta\alpha}^{L}(\mathcal{S}(M_{1},M_{2}))$ and

$T_{\alpha\beta}^{L}(\textup{SC}(\gamma))=\textup{SC}(T_{\alpha\beta}^{L}(\gamma)).$

###### REMARK.

As noted in Remark 2.1.5, the notion of inclusion in a $\tau$-invariant for
elements of $\mathcal{S}(M_{1},M_{2})$ is not defined as inclusion in a set,
but rather as formal statement. We remedy this as follows. Let
$M\subset\mathbb{N}^{*}$ be a finite set. Define $\tau(M)$ as a subset of
$\\{\alpha_{1}^{\prime}\\}\cup\\{\alpha_{i}\;|\;1<i<\infty\\}$ by

  1. (1)

$\alpha_{1}^{\prime}\in\tau(M)$ if and only if $\\{1,2\\}\subseteq M$, and

  2. (2)

for $i>1$, $\alpha_{i}\in\tau(M)$ if and only if $\\{i,i+1\\}\subseteq M$.

We then define, say, for $\gamma\in\mathcal{S}(M_{1},M_{2})$,
$\tau^{L}(\gamma)\subseteq\tau(M_{1})$, adapting the previous definitions to
this context in the obvious way. Strictly speaking, the $\tau$-invariants of
this paper must consequently be distinguished from those in [4] and [5], but
we will neglect doing so.

###### 4.3.3. DEFINITION.

Let $\mathbf{T}\in\mathcal{T}(M)$ and suppose $\\{1,2\\}\subseteq M$. We say
that $\alpha_{1}^{\prime}\in\tau^{L}(\mathbf{T})$ if either:

  1. (1)

$P(1,\mathbf{T})=\\{S_{1,1},S_{2,1}\\}$ and
$P(2,\mathbf{T})=\\{S_{1,2},S_{2,2}\\}$,

  2. (2)

$P(1,\mathbf{T})=\\{S_{1,1},S_{1,2}\\}$ and
$P(2,\mathbf{T})=\\{S_{2,1},S_{3,1}\\}$, or

  3. (3)

$P(1,\mathbf{T})=\\{S_{1,1},S_{2,1}\\}$ and
$P(2,\mathbf{T})=\\{S_{3,1},S_{4,1}\\}$.

Otherwise we say that $\alpha_{1}^{\prime}\notin\tau^{L}(\mathbf{T})$.

###### REMARK.

  1. (1)

With $\mathbf{T}$ as above, we have $\alpha_{1}^{\prime}\in\tau(\mathbf{T})$
if and only if $\alpha_{1}^{\prime}\notin\tau(^{t}\mathbf{T})$.

  2. (2)

Proposition 3.1.4 holds in this setting; that is, the $\tau$-invariant of a
domino tableau is left unchanged by moving through either an open cycle or a
closed cycle of length greater than two.

We will view the map $A$ of Definition 1.2.1 as a bijection from
$\mathcal{S}(M_{1},M_{2})$ to $\mathcal{T}_{D}(M_{1},M_{2})$ and write
$A(\gamma)=(\mathbf{L}(\gamma),\mathbf{R}(\gamma))$. The following is obvious
by inspection:

###### 4.3.4. PROPOSITION.

Suppose $\gamma\in\mathcal{S}(M_{1},M_{2})$ and $\\{1,2\\}\subseteq M_{1}.$
Then $\alpha_{1}^{\prime}\in\tau^{L}(\gamma)$ if and only if
$\alpha_{1}^{\prime}\in\tau(\mathbf{L}(\gamma))$ and
$\alpha_{1}^{\prime}\in\tau^{R}(\gamma)$ if and only if
$\alpha_{1}^{\prime}\in\tau(\mathbf{R}(\gamma))$

When $\\{\alpha,\beta\\}=\\{\alpha_{i},\alpha_{i+1}\\}$ or
$\\{\alpha_{1}^{\prime},\alpha_{3}\\}$, we define the domains
$D_{\alpha\beta}^{L}(\mathcal{T}_{D}(M_{1},M_{2}))$ and
$D_{\alpha\beta}^{R}(\mathcal{T}_{D}(M_{1},M_{2}))$ as in Definition 2.1.11.
When $\\{\alpha,\beta\\}=\\{\alpha_{i},\alpha_{i+1}\\}$, we also adopt the
corresponding definitions of $T_{\alpha\beta}^{L}$ and $T_{\alpha\beta}^{R}.$
When $\\{\alpha,\beta\\}=\\{\alpha_{1}^{\prime},\alpha_{3}\\}$, the definition
is more intricate, as below:

###### 4.3.5. DEFINITION.

Suppose $\\{\alpha,\beta\\}=\\{\alpha_{1}^{\prime},\alpha_{3}\\}$. We define

$T_{\alpha\beta}^{L}:D_{\alpha\beta}^{L}(\mathcal{T}_{D}(M_{1},M_{2}))\longrightarrow
D_{\beta\alpha}^{L}(\mathcal{T}_{D}(M_{1},M_{2}))$

via several cases. Let $(\mathbf{T}_{1},\mathbf{T}_{2})\in
D_{\alpha\beta}^{L}(\mathcal{T}_{D}(M_{1},M_{2})).$

  1. (1)

Suppose $S_{2,2}\notin P(2,\mathbf{T}_{1})\cup P(3,\mathbf{T}_{1})$. Then we
set
$T_{\alpha\beta}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))=(\textup{In}(2,3;\mathbf{T}_{1}),\mathbf{T}_{2}).$

  2. (2)

Suppose $P(1,\mathbf{T}_{1})=\\{S_{1,1},S_{2,1}\\}$,
$P(2,\mathbf{T}_{1})=\\{S_{1,2},S_{2,2}\\}$, and
$P(3,\mathbf{T}_{1})=\\{S_{1,3},S_{1,4}\\}$. Set
$(\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2})=\mathbf{E}((\mathbf{T}_{1},\mathbf{T}_{2}),ec(3,\mathbf{T}_{1};\mathbf{T}_{2}),L).$
Let
$\mathbf{T}_{1}^{\prime}=(\overline{\mathbf{T}}_{1}\setminus\\{(2,S_{2,2}),(3,S_{1,3})\\})\cup\\{(2,S_{1,3}),(3,S_{2,2})\\}.$
We then define

$T_{\alpha\beta}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))=(\mathbf{T}_{1}^{\prime},\overline{\mathbf{T}}_{2}).$

  3. (3)

Suppose $P(1,\mathbf{T}_{1})=\\{S_{1,1},S_{2,1}\\}$,
$P(2,\mathbf{T}_{1})=\\{S_{1,2},S_{2,2}\\}$, and
$P(3,\mathbf{T}_{1})=\\{S_{1,3},S_{2,3}\\}$. Set
$\mathbf{T}_{1}^{\prime}=(\mathbf{T}_{1}\setminus\\{(2,S_{2,2}),(3,S_{1,3})\\})\cup\\{(2,S_{1,3}),(3,S_{2,2})\\}.$
We then define

$T_{\alpha\beta}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))=\mathbf{E}((\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}),ec(3,\mathbf{T}_{1}^{\prime};\mathbf{T}_{2}),L).$

  4. (4)

In any of the settings not listed among (1)-(3),
$T_{\alpha\beta}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$ is defined by the rules
$T_{\alpha\beta}^{L}=(T_{\beta\alpha}^{L})^{-1}$ and
${}^{t}(T_{\alpha\beta}^{L}((\mathbf{T}_{1},\mathbf{T}_{2})))=T_{\beta\alpha}^{L}(^{t}(\mathbf{T}_{1},\mathbf{T}_{2}))$.

The operators $T_{\alpha\beta}^{R}$ are defined in an analogous manner.

###### REMARK.

An alternate description of $T_{\alpha\beta}((\mathbf{T}_{1},\mathbf{T}_{2}))$
in cases (2) and (3) can be given as follows. Let

$H_{1}=$ $1$ $2$ $3$ and $H_{2}=$ $1$ $2$ $3$

In case (2), set $\tilde{\mathbf{T}}_{1}=(\mathbf{T}_{1}\setminus H_{1})\cup
H_{2}$. Then
$T_{\alpha\beta}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))=\mathbf{E}((\tilde{\mathbf{T}}_{1},\mathbf{T}_{2}),ec(3,\tilde{\mathbf{T}}_{1};\mathbf{T}_{2}),L).$
In case (3), set
$(\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2})=\mathbf{E}((\mathbf{T}_{1},\mathbf{T}_{2}),ec(3,\mathbf{T}_{1};\mathbf{T}_{2}),L)$.
Then

$T_{\alpha\beta}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))=((\overline{\mathbf{T}}_{1}\setminus
H_{1})\cup H_{2},\overline{\mathbf{T}}_{2}).$

###### 4.3.6. PROPOSITION.

Let $(\mathbf{T}_{1},\mathbf{T}_{2})\in
D_{\alpha\beta}^{L}(\mathcal{T}_{D}(M_{1},M_{2})).$ We have

$(\mathbf{T}_{1},\mathbf{T}_{2})\in\mathcal{T}^{\prime}(M_{1},M_{2})\text{\hskip
7.22743pt if and only if \hskip 7.22743pt
}T_{\alpha\beta}^{L}(\mathbf{T}_{1},\mathbf{T}_{2})\in\mathcal{T}^{\prime}(M_{1},M_{2}).$

The corresponding statement holds when $L$ is replaced by $R$.

###### Proof.

If $\\{\alpha,\beta\\}=\\{\alpha_{i},\alpha_{i+1}\\}$ for some $i\geq 2$, then
this is clear from the definitions. If on the other hand
$\\{\alpha,\beta\\}=\\{\alpha_{1}^{\prime},\alpha_{3}\\}$, we also can use
Proposition 4.2.4. ∎

###### 4.3.7. LEMMA.

Let $\gamma\in\mathcal{S}(M_{1},M_{2})$ and suppose that $1\in M_{1}$. Let
$\gamma^{\prime}=\textup{SC}^{L}(1;\gamma)$. Then

$A(\gamma^{\prime})=\mathbf{E}(A(\gamma),ec(1,\mathbf{L}(\gamma);\mathbf{R}(\gamma)),L).$

###### Proof.

Let $k$ and $\epsilon$ be such that $(1,k,\epsilon)\in\gamma$. Assume first
that $k=\sup M_{1}$. Let
$\overline{\gamma}=\gamma\setminus\\{(1,k,\epsilon)\\}$, so that
$\gamma^{\prime}=\overline{\gamma}\cup\\{(1,k,-\epsilon\\}$. If
$\overline{\gamma}=\varnothing$, then the lemma is obvious, so assume
otherwise.

Consider a nonempty $\mathbf{T}\in\mathcal{T}(M)$ for some $M$, with $1\notin
M$. If we set $(\mathbf{T}_{1},P_{1})=\alpha((\mathbf{T},1,1))$ and
$(\mathbf{T}_{2},P_{2})=\alpha((\mathbf{T},1,-1))$, and if $l\in M$ is such
that $(l,S_{1,1})\in\mathbf{T}$, then we have
$P(1,\mathbf{T}_{1})=\\{S_{1,1},S_{1,2}\\}$,
$P(l,\mathbf{T}_{1})=\\{S_{2,1},S_{2,2}\\}$,
$P(1,\mathbf{T}_{2})=\\{S_{1,1},S_{2,1}\\}$, and
$P(l,\mathbf{T}_{2})=\\{S_{1,2},S_{1,2}\\}$. For $r\in M\setminus\\{l\\}$ we
have $P(r,\mathbf{T}_{1})=P(r,\mathbf{T}_{2})$, and thus in particular
$P_{1}=P_{2}$. Applying this observation with $\mathbf{L}(\overline{\gamma})$
substituted for $\mathbf{T}$, we find that $c(1,\mathbf{L}(\gamma))$ is a
closed cycle consisting of two elements, that
$\mathbf{L}(\gamma^{\prime})=\mathbf{E}(\mathbf{L}(\gamma),c(1,\mathbf{L}(\gamma))),$
and that $\mathbf{R}(\gamma^{\prime})=\mathbf{R}(\gamma)$, as desired.

Now suppose that $k\neq\sup M_{1}$. Then the lemma follows from the previous
case by repeated application of Proposition 2.3.2 (b).

∎

###### 4.3.8. THEOREM.

Let
$\\{\alpha,\beta\\}=\\{\alpha^{\prime}_{1},\alpha_{3}\\}\subseteq\Pi^{\prime}$.

  1. (1)

Suppose $\gamma\in D_{\alpha\beta}^{L}(\mathcal{S}(M_{1},M_{2})).$ Then
$A(T_{\alpha\beta}^{L}(\gamma))=T_{\alpha\beta}^{L}(A(\gamma))$.

  2. (2)

Suppose $\gamma\in D_{\alpha\beta}^{R}(\mathcal{S}(M_{1},M_{2})).$ Then
$A(T_{\alpha\beta}^{R}(\gamma))=T_{\alpha\beta}^{R}(A(\gamma))$.

###### REMARK.

When $\\{\alpha,\beta\\}=\\{\alpha_{i},\alpha_{i+1}\\}$ with $i\geq 2$, the
corresponding theorem was proved as Theorem 2.1.19

Proof of Theorem 4.3.8.

As usual, it suffices to prove the first part. Since $T_{\alpha\beta}^{L}$ and
$T_{\beta\alpha}^{L}$ are inverses, we may assume that $\alpha=\alpha_{3}$.
For $j\in M_{1}$, define $k_{j}$ and $\epsilon_{j}$ by
$(j,k_{j},\epsilon_{j})\in\gamma$. Set
$\overline{\gamma}=\gamma\setminus\\{(j,k_{j},\epsilon_{j})\;|\;j\in\\{1,2,3\\}\\}$.
Let $d<e<f$ be such that $\\{d,e,f\\}=\\{k_{1},k_{2},k_{3}\\}.$ Set
$\gamma^{\prime}=T_{\alpha\beta}^{L}(\gamma)$. Then one of the following ten
cases holds:

  1. (1)

$\gamma\supseteq\\{(1,f,-1),(2,e,-1),(3,d,\epsilon)\\}$ and
$\gamma^{\prime}=\overline{\gamma}\cup\\{(1,e,1),(2,f,1),(3,d,\epsilon)\\}.$

  2. (2)

$\gamma\supseteq\\{(1,e,-1),(2,d,\epsilon),(3,f,1)\\}$ and
$\gamma^{\prime}=\overline{\gamma}\cup\\{(1,e,-1),(2,f,1),(3,d,\epsilon)\\}.$

  3. (3)

$\gamma\supseteq\\{(1,f,-1),(2,d,-1),(3,e,1)\\}$ and
$\gamma^{\prime}=\overline{\gamma}\cup\\{(1,d,1),(2,f,1),(3,e,1)\\}.$

  4. (4)

$\gamma\supseteq\\{(1,f,-1),(2,d,1),(3,e,1)\\}$ and
$\gamma^{\prime}=\overline{\gamma}\cup\\{(1,d,-1),(2,f,1),(3,e,1)\\}.$

  5. (5)

$\gamma\supseteq\\{(1,d,\epsilon),(2,f,-1),(3,e,1)\\}$ and
$\gamma^{\prime}=\overline{\gamma}\cup\\{(1,d,\epsilon),(2,e,1),(3,f,-1)\\}.$

Cases (6)-(10) are obtained from (1)-(5) by replacing $\gamma$ with
$\textup{SC}(\gamma^{\prime})$. By Remarks 2.1.6-(2) and 4.3.2-(2) as well as
Definition 4.3.5 (4), we need only consider cases (1) through (5).

In case (1), set
$\gamma_{1}=\overline{\gamma}\cup\\{(1,e,-1),(2,f,1),(3,d,\epsilon)\\}$. We
observe directly that $A(\gamma)$ satisfies the hypothesis of case (3) of
Definition 4.3.5, that
$\mathbf{L}(\gamma_{1})=(\mathbf{L}(\gamma)\setminus\\{(2,S_{2,2}),(3,S_{1,3})\\})\cup\\{(2,S_{1,3}),(3,S_{2,2})\\}$,
and that $\mathbf{R}(\gamma_{1})=\mathbf{R}(\gamma)$. On the other hand, Lemma
4.3.7 says that

$A(\gamma^{\prime})=\mathbf{E}(A(\gamma_{1}),ec(1,\mathbf{L}(\gamma_{1});\mathbf{R}(\gamma_{1})),L).$

Since $c(3,\mathbf{L}(\gamma_{1}))=c(1,\mathbf{L}(\gamma_{1}))$, case (1) is
complete.

In case (2), set
$\gamma_{1}=\overline{\gamma}\cup\\{(1,e,1),(2,f,1),(3,d,\epsilon)\\}$.
Adopting the notation from Remark 4.3.5, we note that
$H_{1}\subseteq\mathbf{L}(\gamma)$,
$\mathbf{L}(\gamma_{1})=(\mathbf{L}(\gamma)\setminus H_{1})\cup H_{2}$, and
that $\mathbf{R}(\gamma_{1})=\mathbf{R}(\gamma)$. As in case (1), we have
$A(\gamma^{\prime})=\mathbf{E}(A(\gamma_{1}),ec(3,\mathbf{L}(\gamma_{1});\mathbf{R}(\gamma_{1})),L)$
and the rest of the case follows by Remark 4.3.5.

The proof of case (3) follows the outline of case (1), provided that we let
$\gamma_{1}=\overline{\gamma}\cup\\{(1,d,-1),(2,f,1),(3,e,1)\\}$, and case (4)
follows as case (2), letting
$\gamma_{1}=\overline{\gamma}\cup\\{(1,d,1),(2,f,1),(3,e,1)\\}$. In case (5),
note that $A(\gamma)$ falls within case (1) of Definition 4.3.5,
$\mathbf{L}(\gamma^{\prime})=\textup{In}(2,3;\mathbf{L}(\gamma))$, and
$\mathbf{R}(\gamma^{\prime})=\mathbf{R}(\gamma)$.

∎

##

We now define a family of operators that extend the generalized
$\tau$-invariant to the type $D$ setting. They were originally introduced in
[6] in the context of primitive ideals, but the construction therein can be
replicated on any set where one has defined a $\tau$-invariant and operators
$T_{\alpha\beta}$ based on a branched Dynkin diagram.

The first part of this section describes the action of the operators in the
settings of signed permutations and pairs of domino tableaux. The second shows
that these two definitions are compatible by verifying operators on signed
permutations and domino tableaux commute with the domino Robinson-Schensted
map $A$. Since in this paper most of the sets in question have left and right
$\tau$-invariants as well as left and right operators $T_{\alpha\beta}$, we
will similarly speak of objects introduced here as being of left or right
type. We begin by describing sets which will become their domains.

###### 4.4.1. DEFINITION.

Suppose either $X=W^{\prime},\,W^{\prime\prime},\,\mathcal{S}(M_{1},M_{2}),$
or $\mathcal{T}_{D}(M_{1},M_{2})$ with $\\{1,2,3,4\\}\subseteq M_{1}$. Let
$x\in X$ and suppose
$\\{\beta,\gamma,\delta\\}=\\{\alpha_{1}^{\prime},\alpha_{2},\alpha_{4}\\}$.
We say that $x$ is of

  1. (1)

left type $\mathcal{A}_{\beta}$ if
$\\{\beta,\alpha_{3}\\}\subseteq\tau^{L}(x)$,
$\\{\gamma,\delta\\}\cap\tau^{L}(x)=\varnothing$, and
$T_{\gamma\alpha_{3}}^{L}(x)\neq T_{\delta\alpha_{3}}^{L}(x),$

  2. (2)

left type $\mathcal{B}_{\beta}$ if $\\{\gamma,\delta\\}\subseteq\tau^{L}(x)$,
$\\{\beta,\alpha_{3}\\}\cap\tau^{L}(x)=\varnothing$, and
$T_{\alpha_{3}\gamma}^{L}(x)\neq T_{\alpha_{3}\delta}^{L}(x),$

  3. (3)

left type $\mathcal{C}$ if $\\{\beta,\gamma,\delta\\}\subseteq\tau^{L}(x)$,
$\alpha_{3}\notin\tau^{L}(x)$, and
$T_{\alpha_{3}\beta}^{L}(x)=T_{\alpha_{3}\gamma}^{L}(x)=T_{\alpha_{3}\delta}^{L}(x),$

  4. (4)

left type $\mathcal{D}$ if $\alpha_{3}\in\tau^{L}(x)$,
$\\{\beta,\gamma,\delta\\}\cap\tau^{L}(x)=\varnothing$, and
$T_{\beta\alpha_{3}}^{L}(x)=T_{\gamma\alpha_{3}}^{L}(x)=T_{\delta\alpha_{3}}^{L}(x),$

  5. (5)

left type $8$ if $x$ is of left type $\mathcal{A}_{\beta}$ or
$\mathcal{B}_{\beta}$ for some
$\beta\in\\{\alpha_{1}^{\prime},\alpha_{2},\alpha_{4}\\}$, or it is of left
type $\mathcal{C}$ or $\mathcal{D}$.

We define right types $\mathcal{A}_{\beta}$, $\mathcal{B}_{\beta}$,
$\mathcal{C}$, $\mathcal{D}$, and $8$ analogously whenever
$\\{1,2,3,4\\}\subseteq M_{2}$.

###### REMARK.

The following relationships also hold with right and left interchanged.

  1. (1)

The element $x$ is of left type $\mathcal{C}$ if and only if
$T^{L}_{\alpha_{3}\beta}(x)$ is of left type $\mathcal{D}$. Also, $x$ is of
left type $\mathcal{A}_{\beta}$ if and only if $T^{L}_{\gamma\alpha_{3}}(x)$
is of left type $\mathcal{B}_{\delta}$.

  2. (2)

If $w\in W^{\prime}$ or $w\in W^{\prime\prime}$, then $w$ is of left type
$\mathcal{A}_{\beta}$ if and only if $\delta(w)$ is of right type
$\mathcal{A}_{\beta}$, and similarly for the other types. This follows from
Remarks 2.1.7 (1) and 4.3.2 (1).

  3. (3)

If $\gamma\in\mathcal{S}(M_{1},M_{2})$, then $\gamma$ is of left type
$\mathcal{A}_{\beta}$ if and only if $A(\gamma)$ is of left type
$\mathcal{A}_{\beta}$, and similarly for the other types. This follows from
Propositions 2.1.18, 4.3.4, and Theorems 2.1.19 and 4.3.8.

In [6], type 8 modules are grouped into sets of ten or fourteen. Each set
consists of either one module each of type $\mathcal{A}_{\beta}$ and
$\mathcal{B}_{\beta}$ for
$\beta\in\\{\alpha_{1}^{\prime},\alpha_{2},\alpha_{4}\\}$ and two each of
types $\mathcal{C}$ and $\mathcal{D}$, or two modules each of the types
$\mathcal{A}_{\beta}$ and $\mathcal{B}_{\beta}$ for
$\beta\in\\{\alpha_{1}^{\prime},\alpha_{2},\alpha_{4}\\}$, one module of type
$\mathcal{C}$, and one module of type $\mathcal{D}$. We will implement this
grouping by means of operators, which, to each type $\mathcal{C}$ (resp, type
$\mathcal{D}$) object associate a set consisting of one or two type
$\mathcal{B}_{\beta}$ (resp. $\mathcal{A}_{\beta}$) objects for
$\beta\in\\{\alpha_{1}^{\prime},\alpha_{2},\alpha_{4}\\}$. Similarly, we
define maps from type $\mathcal{A}_{\beta}$ objects to sets of type
$\mathcal{D}$ objects, and from type $\mathcal{B}_{\beta}$ objects to sets of
type $\mathcal{C}$ objects. We now define these correspondences for the sets
considered in Definition 4.4.1.

###### 4.4.2. NOTATION.

Let $\gamma\in\mathcal{S}(M_{1},M_{2})$. If $\\{i-1,i\\}\subseteq M_{1}$, we
set $s_{\alpha_{i}}\gamma=s_{i}\gamma=\textup{In}^{L}(i-1,i;\gamma).$ If
$\\{1,2\\}\subseteq M_{1}$, we define
$s_{\alpha_{1}^{\prime}}\gamma=s_{1}^{\prime}\gamma=\textup{SC}^{L}(1;\textup{SC}^{L}(2;(\textup{In}^{L}(1,2;\gamma)))).$
If on the other hand $\\{i-1,i\\}\subseteq M_{2}$ or $\\{1,2\\}\subseteq
M_{2}$, we define $\gamma s_{\alpha_{i}}$ and $\gamma s_{\alpha_{1}^{\prime}}$
by replacing $L$ with $R$ in the above.

###### REMARK.

We have $\delta(s\gamma)=\delta(\gamma)s$ and $\delta(\gamma
s)=s\delta(\gamma)$ for $s\in\\{s_{1}^{\prime},s_{2},\ldots,s_{n}\\}$.

###### 4.4.3. DEFINITION.

Let $w\in W^{\prime}$ or $W"$ or $\mathcal{S}(M_{1},M_{2})$ and
$\\{\beta,\gamma,\delta\\}=\\{\alpha_{1}^{\prime},\alpha_{2},\alpha_{4}\\}\subseteq\Pi^{\prime}$.

  1. (1)

If $w$ is of left type $\mathcal{A}_{\beta}$, we define

$T_{D\beta}^{L}(w)=\\{y\in\\{s_{\beta}w,s_{\alpha_{3}}s_{\gamma}s_{\delta}w,s_{\alpha_{3}}s_{\delta}s_{\alpha_{3}}w,s_{\alpha_{3}}s_{\gamma}s_{\alpha_{3}}w\\}\;|\;y\text{
is of left type $\mathcal{D}$}\\}.$

  2. (2)

If $w$ is of left type $\mathcal{B}_{\beta}$, we define

$T_{C\beta}^{L}(w)=\\{y\in\\{s_{\beta}w,s_{\alpha_{3}}s_{\gamma}s_{\delta}w,s_{\alpha_{3}}s_{\delta}s_{\alpha_{3}}w,s_{\alpha_{3}}s_{\gamma}s_{\alpha_{3}}w\\}\;|\;y\text{
is of left type $\mathcal{C}$}\\}.$

  3. (3)

If $w$ is of left type $\mathcal{C}$, we define

$T_{\beta
C}^{L}(w)=\\{y\in\\{s_{\beta}w,s_{\alpha_{3}}s_{\gamma}s_{\delta}w,s_{\alpha_{3}}s_{\delta}s_{\alpha_{3}}w,s_{\alpha_{3}}s_{\gamma}s_{\alpha_{3}}w\\}\;|\;y\text{
is of left type $\mathcal{\mathcal{B}_{\beta}}$}\\}.$

  4. (4)

If $w$ is of left type $\mathcal{D}$, we define

$T_{\beta
D}^{L}(w)=\\{y\in\\{s_{\beta}w,s_{\alpha_{3}}s_{\gamma}s_{\delta}w,s_{\alpha_{3}}s_{\delta}s_{\alpha_{3}}w,s_{\alpha_{3}}s_{\gamma}s_{\alpha_{3}}w\\}\;|\;y\text{
is of left type $\mathcal{\mathcal{A}_{\beta}}$}\\}.$

  5. (5)

If $w$ is of right type $\mathcal{\mathcal{A}_{\beta}}$, we define

$T_{D\beta}^{R}(w)=\\{y\in\\{ws_{\beta},ws_{\alpha_{3}}s_{\gamma}s_{\delta},ws_{\alpha_{3}}s_{\delta}s_{\alpha_{3}},ws_{\alpha_{3}}s_{\gamma}s_{\alpha_{3}}\\}\;|\;y\text{
is of right type $\mathcal{D}$}\\}$

and proceed to define $T_{C\beta}^{R}(w)$, $T_{\beta D}^{R}(w)$ and $T_{\beta
C}^{R}(w)$ analogously.

###### REMARK.

The following also hold with right and left interchanged.

  1. (1)

If $w\in W^{\prime}$ is of left type $\mathcal{A}_{\beta}$, we have
$\delta(T_{D\beta}^{L}(w))=T_{D\beta}^{R}((\delta(w))$, and similarly for the
other types.

  2. (2)

Let $\gamma\in\mathcal{S}(M_{1},M_{2})$. Then $\gamma$ is of left type
$\mathcal{C}$ if and only if $\textup{SC}(\gamma)$ is of left type
$\mathcal{D}$, in which case $\textup{SC}(T_{\beta C}^{L}(\gamma))=T_{\beta
D}(\textup{SC}(\gamma))$. Similarly, $\gamma$ is of left type
$\mathcal{A}_{\beta}$ if and only if $\textup{SC}(\gamma)$ is of left type
$\mathcal{B}_{\beta}$, in which case
$\textup{SC}(T_{D\beta}^{L}(\gamma))=T_{C\beta}(\textup{SC}(\gamma))$.

  3. (3)

Let $\gamma\in\mathcal{S}(M_{1},M_{2})$. If $\gamma$ is of left type
$\mathcal{A}_{\beta}$, then $\gamma\in\mathcal{S}^{\prime}(M_{1},M_{2})$ if
and only if
$T_{D\beta}^{L}(\gamma)\subseteq\mathcal{S}^{\prime}(M_{1},M_{2})$, and
similarly for the other types.

###### 4.4.4. PROPOSITION.

Adopt the notation of Definition 4.4.3. The following statements also hold
with left replaced by right. The first two also hold with $C$ and $D$
interchanged.

  1. (1)

The sets $T_{\beta C}^{L}(w)$ and $T_{C\beta}^{L}(w)$ consist of one or two
elements. Furthermore:

    1. (i)

If $T_{\beta C}^{L}(w)=\\{w^{\prime}\\}$, then
$T_{C\beta}^{L}(w^{\prime})=\\{w,w"\\}$ with $w"\neq w^{\prime}$. In this
case, we also have $T_{\beta C}^{L}(w")=\\{w^{\prime}\\}$.

    2. (ii)

If $T_{\beta C}^{L}(w)=\\{w_{1},w_{2}\\}$ with $w_{1}\neq w_{2}$, then
$T_{C\beta}^{L}(w_{1})=T_{C\beta}^{L}(w_{2})=\\{w\\}$.

  2. (2)

If $w$ is of left type $\mathcal{C}$, then $T_{\gamma
D}^{L}(T_{\alpha_{3}\beta}^{L}(w))=T_{\alpha_{3}\gamma}^{L}(w)$. In
particular,

$|T_{\beta C}^{L}(w)|=|T_{\gamma C}^{L}(w)|=|T_{\delta C}^{L}(w)|.$

  3. (3)

If $w$ is of left type $\mathcal{C}$ and if $T_{\beta
C}^{L}(w)=\\{w_{1},w_{2}\\}$, then

$w_{2}=(T_{\delta\alpha_{3}}^{L}\circ T_{\alpha_{3}\beta}^{L}\circ
T_{\gamma\alpha_{3}}^{L}\circ T_{\alpha_{3}\delta}^{L}\circ
T_{\beta\alpha_{3}}^{L}\circ T_{\alpha_{3}\gamma}^{L})(w_{1}).$

If $w$ is of left type $\mathcal{D}$ and if $T_{\beta
D}^{L}(w)=\\{w_{1},w_{2}\\}$, then

$w_{2}=(T_{\alpha_{3}\delta}^{L}\circ T_{\beta\alpha_{3}}^{L}\circ
T_{\alpha_{3}\gamma}^{L}\circ T_{\delta\alpha_{3}}^{L}\circ
T_{\alpha_{3}\beta}^{L}\circ T_{\gamma\alpha_{3}}^{L})(w_{1}).$

###### Proof.

For $W^{\prime}$ when $n=4$, these statements can be verified directly by
explicit case-by-case analysis. For $n>4$, they can be deduced from the $n=4$
case using [2, Exercise 3, §1]. Alternately, they follow from Theorem 2.15 and
Proposition 4.4 of [6], regarding these results as statements about
irreducible highest weight modules. The statements for $\mathcal{S}(n,n)$
follow from this, using Remarks 2.1.7 (1) and 4.4.2, and therefore for any
$\mathcal{S}(M_{1},M_{2})$ by means of an obvious bijection. ∎

We now proceed to describe the action of the operators defined above on the
tableaux pairs $\mathcal{T}_{D}(M_{1},M_{2})$. Our first task is to simplify
the characterization of left types in the setting of domino tableaux.

###### 4.4.5. NOTATION.

We define certain subsets of $\mathcal{F}$ using the natural identification
with Young diagrams. For $X\in\\{A,B,C,D\\}$, let

$F_{X}^{1}=$ $F_{X}^{2}=$

When $X\in\\{A,B\\}$, let

$F_{X}^{3}=$ $F_{X}^{4}=$

And when $X\in\\{C,D\\}$, let

$F_{X}^{3}=$ $F_{X}^{4}=$

We use the above shapes to replace the final condition in the definition of
each of the left types. The following can be verified easily from the
definitions:

###### 4.4.6. PROPOSITION.

Let $(\mathbf{T}_{1},\mathbf{T}_{2})\in\mathcal{T}_{D}(M_{1},M_{2})$ and
suppose $\\{1,2,3,4\\}\subseteq M_{1}$. Then $(\mathbf{T}_{1},\mathbf{T}_{2})$
is of left type $\mathcal{A}_{\alpha^{\prime}_{1}}$ if and only if
$\\{\alpha_{1}^{\prime},\alpha_{3}\\}\subseteq\tau(\mathbf{T}_{1})$,
$\\{\alpha_{2},\alpha_{4}\\}\cap\tau(\mathbf{T}_{1})=\varnothing$, and
$\bigcup_{k=1}^{4}P(k,\mathbf{T}_{1})\in\\{F_{A}^{i}\;|\;1\leq i\leq 4\\}$.
The obvious analogues of this statement hold for each of the other types.

In particular, the proposition implies that $\mathbf{T}_{1}$ determines the
left type of the pair $(\mathbf{T}_{1},\mathbf{T}_{2})$, allowing us to abuse
terminology and speak of an individual domino tableau as being of a particular
left type itself. It is clear that whether a domino tableau is of a particular
left type can be determined by its first four dominos. Furthermore, it is easy
to check that specifying a left type for a domino tableau in
$\mathcal{T}_{D}(4)$ together with a compatible shape from 4.4.5 determines
the tableau itself uniquely. We exploit this in the following definition.

###### 4.4.7. DEFINITION.

For $1\leq j\leq 4$ and
$\beta\in\\{\alpha^{\prime}_{1},\alpha_{2},\alpha_{4}\\}$ we let

$\displaystyle A_{\beta}^{j}\text{ be unique tableau of type }$
$\displaystyle\mathcal{A}_{\beta}\text{ and
}\textup{Shape}(A^{j}_{\beta})=F^{j}_{A},$ $\displaystyle B_{\beta}^{j}\text{
be unique tableau of type }$ $\displaystyle\mathcal{B}_{\beta}\text{ and
}\textup{Shape}(B^{j}_{\beta})=F^{j}_{B},$ $\displaystyle C^{j}\text{ be
unique tableau of type }$
$\displaystyle\phantom{.}\mathcal{C}\phantom{.}\text{ and
}\textup{Shape}(C^{j})=F^{j}_{C},$ $\displaystyle\text{ and \hskip
8.61108pt}D^{j}\text{ be unique tableau of type }$
$\displaystyle\phantom{.}\mathcal{D}\phantom{.}\text{ and
}\textup{Shape}(D^{j})=F^{j}_{D}.$

For completeness, we list these tableaux explicitly in Figure 4.1.

$A_{1}^{1}=$ $1$ $2$ $4$ $3$ $A_{1}^{2}=$ $1$ $2$ $4$ $3$ $A_{1}^{3}=$ $1$ $2$
$4$ $3$ $A_{1}^{4}=$ $1$ $2$ $4$ $3$

$A_{2}^{1}=$ $1$ $4$ $2$ $3$ $A_{2}^{2}=$ $1$ $4$ $2$ $3$ $A_{2}^{3}=$ $1$ $4$
$2$ $3$ $A_{2}^{4}=$ $1$ $4$ $2$ $3$

$A_{4}^{1}=$ $1$ $2$ $3$ $4$ $A_{4}^{2}=$ $1$ $2$ $3$ $4$ $A_{4}^{3}=$ $1$ $2$
$3$ $4$ $A_{4}^{4}=$ $1$ $2$ $3$ $4$

LABEL:

$B_{1}^{1}=$ $1$ $3$ $2$ $4$ $B_{1}^{2}=$ $1$ $3$ $2$ $4$ $B_{1}^{3}=$ $1$ $3$
$2$ $4$ $B_{1}^{4}=$ $1$ $3$ $2$ $4$

$B_{2}^{1}=$ $1$ $2$ $3$ $4$ $B_{2}^{2}=$ $1$ $2$ $3$ $4$ $B_{2}^{3}=$ $1$ $2$
$3$ $4$ $B_{2}^{4}=$ $1$ $2$ $3$ $4$

$B_{4}^{1}=$ $1$ $3$ $4$ $2$ $B_{4}^{2}=$ $1$ $4$ $2$ $3$ $B_{4}^{3}=$ $1$ $3$
$4$ $2$ $B_{4}^{4}=$ $1$ $4$ $2$ $3$

$C^{1}=$ $1$ $3$ $4$ $2$ $C^{2}=$ $1$ $3$ $2$ $4$ $C^{3}=$ $1$ $3$ $4$ $2$
$C^{4}=$ $1$ $3$ $2$ $4$

$D^{1}=$ $1$ $2$ $4$ $3$ $D^{2}=$ $1$ $2$ $3$ $4$ $D^{3}=$ $1$ $2$ $4$ $3$
$D^{4}=$ $1$ $2$ $3$ $4$

Figure 4.1: Tableaux of Definition 4.4.7. Condensing notation, we write
$A_{1}^{j}$ in lieu of $A_{\alpha_{1}^{\prime}}^{j}$ and similarly for the
other types and roots.

###### 4.4.8. DEFINITION.

Tableaux $X$ and $Y$ taken from among those enumerated in Definition 4.4.7
will be called a replaceable pair if for some $j\in\\{1,2\\}$, we have
$X=C^{j}$ and $Y\in\\{B_{1}^{j},B_{2}^{j},B_{4}^{j}\\}$, or $X=D^{j}$ and
$Y\in\\{A_{1}^{j},A_{2}^{j},A_{4}^{j}\\}$, or this holds with $X$ and $Y$
interchanged. If $X\subseteq\mathbf{T}$ and $X$ together with $Y$ form a
replaceable pair, we will write
$\textup{Re}(X,Y;\mathbf{T})=(\mathbf{T}\setminus X)\cup Y$.

###### 4.4.9. LEMMA.

Consider $(\mathbf{T}_{1},\mathbf{T}_{2})\in\mathcal{T}_{D}(M_{1},M_{2})$. If
$X$ and $Y$ form a replaceable pair, $X\subseteq\mathbf{T}_{1},$ and
$\mathbf{T}_{1}^{\prime}=\textup{Re}(X,Y;\mathbf{T}_{1}),$ then $3\in
ec(4,\mathbf{T}_{1};\mathbf{T}_{2})\text{ if and only if }3\notin
ec(4,\mathbf{T}_{1}^{\prime};\mathbf{T}_{2}).$

###### Proof.

This can be observed directly when $|M_{1}|=4$. Otherwise, the argument is by
induction using Proposition 2.3.3 (b), and as in the proof of Remark 2.3.4. ∎

We are ready to describe the action of the operators of this section on pairs
of domino tableaux.

###### 4.4.10. DEFINITION.

Let $(\mathbf{T}_{1},\mathbf{T}_{2})\in\mathcal{T}_{D}(M_{1},M_{2})$ and
suppose it is of left type $\mathcal{C}$. Let $j$ be such that
$C^{j}\subseteq\mathbf{T}_{1}$. If $j\in\\{1,2\\}$, write
$\mathbf{T}_{1}^{\prime}=\textup{Re}(C^{j},B^{j}_{1};\mathbf{T}_{1})$ and set
$c_{k}=ec(k,\mathbf{T}_{1}^{\prime};\mathbf{T}_{2})$. There are two main
cases:

  1. (1)

Suppose $4\in ec(3,\mathbf{T}_{1};\mathbf{T}_{2})$ so that $j\in\\{1,2\\}$. By
Lemma 4.4.9, $c_{3}\neq c_{4}$. We define

$T_{\alpha^{\prime}_{1}C}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))=\left\\{\begin{array}[]{lr}\big{\\{}(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2})\;,\;\mathbf{E}((\mathbf{T}_{1}^{\prime},\mathbf{T}_{2});\\{c_{3},c_{4}\\},L)\big{\\}}&\text{if
}n_{h}(C^{j})=n_{h}(B_{1}^{j}),\vspace{.1in}\\\
\big{\\{}\mathbf{E}((\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}),c_{3},L)\;,\;\mathbf{E}((\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}),c_{4},L)\big{\\}}&\text{if
}n_{h}(C^{j})\neq n_{h}(B_{1}^{j}).\end{array}\right.$

  2. (2)

Suppose $4\notin ec(3,\mathbf{T}_{1};\mathbf{T}_{2}).$ If $j\in\\{1,2\\}$, we
define

$T_{\alpha^{\prime}_{1}C}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))=\left\\{\begin{array}[]{lr}\big{\\{}(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2})\big{\\}}&\phantom{XXXXXXXXX}\text{if
}n_{h}(C^{j})=n_{h}(B_{1}^{j}),\vspace{.1in}\\\
\big{\\{}\mathbf{E}((\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}),c_{3},L)\big{\\}}&\text{if
}n_{h}(C^{j})\neq n_{h}(B_{1}^{j}).\end{array}\right.$

If $j\in\\{3,4\\},$ we let $k\in\\{1,2\\}$ be such that $n_{h}(C^{k})\neq
n_{h}(B_{1}^{k})$, and we let $l\in\\{3,4\\}$ be such that
$\mathbf{E}(\mathbf{C}^{j},c(l,\mathbf{C}^{j}))=\mathbf{C}^{k}$, where
$\mathbf{C}^{j}=(C^{j},\phi_{D})$. We set
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})=\mathbf{E}((\mathbf{T}_{1},\mathbf{T}_{2}),ec(l,\mathbf{T}_{1};\mathbf{T}_{2}),L)$,
and define

$T_{\alpha^{\prime}_{1}C}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))=\big{\\{}(\textup{Re}(C^{k},B_{1}^{k};\mathbf{T}_{1}^{\prime}),\mathbf{T}_{2}^{\prime})\big{\\}}.$

We extend the above definition to all operators of the form $T^{L}_{\alpha
X}$, $T^{L}_{X\alpha}$, $T^{R}_{\alpha X}$, $T^{R}_{X\alpha}$ using the
obvious analogues.

###### REMARK.

We make the following observations:

  1. (1)

The statements of Proposition 4.4.4 hold for the maps defined in Definition
4.4.10.

  2. (2)

Let $Y\in\mathcal{T}_{D}(M_{1},M_{2})$. Then $Y$ is of left type $\mathcal{C}$
if and only if ${}^{t}Y$ is of left type $\mathcal{D}$. If $Y$ is of left type
$\mathcal{C}$, then ${}^{t}(T^{L}_{\beta C}(Y))=T^{L}_{\beta D}(^{t}Y)$.
Similarly, if $Y$ is of left type $\mathcal{A}_{\beta}$, then
${}^{t}(T^{L}_{D\beta}(Y))=T^{L}_{C\beta}(^{t}Y)$. Similarly with right in
place of left.

  3. (3)

Let $Y\in\mathcal{T}_{D}(M_{1},M_{2})$. If $Y$ is of left type
$\mathcal{A}_{\beta}$, then $Y\in\mathcal{T}^{\prime}_{D}(M_{1},M_{2})$ if and
only if $T^{L}_{D\beta}(Y)\subseteq\mathcal{T}_{D}^{\prime}(M_{1},M_{2})$, and
similarly for the other types. This uses Proposition 4.2.4.

###### 4.4.11. PROPOSITION.

Let $\gamma\in\mathcal{S}(M_{1},M_{2})$ and let
$\beta\in\\{\alpha^{\prime}_{1},\alpha_{2},\alpha_{4}\\}$.

  1. (1)

    1. (a)

If $\gamma$ is of left type $\mathcal{A}_{\beta}$, then
$T^{L}_{D\beta}(A(\gamma))=A(T^{L}_{D\beta}(\gamma))$.

    2. (b)

If $\gamma$ is of left type $\mathcal{B}_{\beta}$, then
$T^{L}_{C\beta}(A(\gamma))=A(T^{L}_{C\beta}(\gamma))$.

    3. (c)

If $\gamma$ is of left type $\mathcal{C}$, then $T^{L}_{\beta
C}(A(\gamma))=A(T^{L}_{\beta C}(\gamma))$.

    4. (d)

If $\gamma$ is of left type $\mathcal{D}$, then $T^{L}_{\beta
D}(A(\gamma))=A(T^{L}_{\beta D}(\gamma))$.

  2. (2)

Analogous statements hold with right in place of left.

###### Proof.

As usual it suffices to prove (1). In fact, using Proposition 4.4.4, Remark
4.4.10 (1), and Theorems 2.1.19 and 4.3.8, it suffices to prove (1)(d).
Further, it is enough to prove that given $\gamma\in\mathcal{S}(M_{1},M_{2})$
of left type $\mathcal{D}$, there is an
$\eta\in\\{\alpha_{1}^{\prime},\alpha_{2},\alpha_{4}\\}$ and a
$\gamma^{\prime}\in T^{L}_{\eta D}(\gamma)$ such that $A(\gamma^{\prime})\in
T^{L}_{\eta D}(A(\gamma))$. So assume $\gamma$ is of left type $\mathcal{D}$.
Define for $j\in M_{1}$, $k_{j}$ and $\epsilon_{j}$ by
$(j,k_{j},\epsilon_{j})\in\gamma$. Let $d<e<f<g$ be such that
$\\{d,e,f,g\\}=\\{k_{1},k_{2},k_{3},k_{4}\\}$. There are sixteen cases. We
list the first eight:

  1. (1)

$\gamma\subseteq\\{(4,d,\epsilon),(1,e,-1),(2,f,1),(3,g,-1)\\}$

  2. (2)

$\gamma\subseteq\\{(4,d,\epsilon),(1,e,-1),(3,f,-1),(2,g,1)\\}$

  3. (3)

$\gamma\subseteq\\{(1,d,-1),(4,e,1),(2,f,1),(3,g,-1)\\}$

  4. (4)

$\gamma\subseteq\\{(1,d,-1),(4,e,-1),(3,f,-1),(2,g,1)\\}$

  5. (5)

$\gamma\subseteq\\{(1,d,-1),(4,e,1),(3,f,-1),(2,g,1)\\}$

  6. (6)

$\gamma\subseteq\\{(1,d,-1),(3,e,-1),(4,f,1),(2,g,1)\\}$

  7. (7)

$\gamma\subseteq\\{(1,d,1),(4,e,1),(3,f,-1),(2,g,1)\\}$

  8. (8)

$\gamma\subseteq\\{(1,d,1),(3,e,-1),(4,f,1),(2,g,1)\\}$

Cases (9)–(16) are obtained from the above by replacing $\gamma$ with
$s_{3}(\textup{SC}(\gamma))=\textup{SC}(s_{3}\gamma)$. Using Remarks
2.1.6-(2), 4.4.3-(2), and 4.4.10-(2), together with the results cited at the
beginning of this proof, we see that is suffices to verfiy the theorem in the
first eight cases. In the first six cases, we observe that $s_{2}\gamma\in
T^{L}_{\alpha_{2}D}(\gamma)$, that $D^{1}\subseteq\mathbf{L}(\gamma)$, that
$\mathbf{L}(s_{2}\gamma)=\textup{Re}(D^{1},A_{2}^{1};\mathbf{L}(\gamma))$, and
that $\mathbf{R}(s_{2}\gamma)=\mathbf{R}(\gamma)$. That is, we have
$A(s_{2}\gamma)\in T^{L}_{\alpha_{2}D}(A(\gamma))$, as desired. In cases (7)
and (8), we observe that $D^{4}\subseteq\mathbf{L}(\gamma)$ and set
$\gamma_{1}=\textup{SC}^{L}(1,\gamma)$. Then by Lemma 4.3.7 and using the fact
that $c(1,\mathbf{L}(\gamma))=c(3,\mathbf{L}(\gamma)),$ we have
$A(\gamma_{1})=\mathbf{E}(A(\gamma),ec(3,\mathbf{L}(\gamma);\mathbf{R}(\gamma)),L).$
In particular, $D^{1}\subseteq\mathbf{L}(\gamma_{1})$. Now we observe that
$s_{1}^{\prime}\gamma\in T^{L}_{\alpha_{1}^{\prime}D}(\gamma),$ that
$\mathbf{L}(s_{1}^{\prime}\gamma)=\textup{Re}(D^{1},A_{1}^{1};\mathbf{L}(\gamma_{1}))$,
and that $\mathbf{R}(s_{1}^{\prime}\gamma)=\mathbf{R}(\gamma_{1})$. That is,
we have $A(s_{1}^{\prime}\gamma)\in T^{L}_{\alpha_{1}^{\prime}D}(A(\gamma))$,
as desired.

∎

##

In this section we describe the interaction of the operators introduced in
this paper and cycles. We note that Propositions 3.1.2 and 3.1.5, which detail
the interaction of operators and cycles for $T_{\alpha\beta}$ when
$\\{\alpha,\beta\\}=\\{\alpha_{i},\alpha_{i+1}\\}$ with $i\geq 2$, extend
directly to type $D$. Hence, we can begin by considering the operators
$T_{\alpha\beta}$ where
$\\{\alpha,\beta\\}=\\{\alpha_{1}^{\prime},\alpha_{3}\\}$.

NOTATION: We will work exclusively with the left versions of all operators and
associated sets, so throughout this section we will drop the superscript $L$.

###### 4.5.1. PROPOSITION.

Let $\mathbf{T}\in\mathcal{T}_{D}(M)$ and suppose that $H_{i}$ or
${}^{t}H_{i}\subseteq\mathbf{T}$ for some $i\in\\{1,2\\}$. Accordingly, let
$\mathbf{T}^{\prime}=(\mathbf{T}\setminus H_{i})\cup H_{3-i}$ or
$(\mathbf{T}\setminus^{t}H_{i})\cup^{t}H_{3-i}$.

  1. (1)

If $k\in M$ but $k\not\in\\{1,2,3\\}$, then
$P^{\prime}(k,\mathbf{T}^{\prime})=P^{\prime}(k,\mathbf{T})$, and in
particular, cycles that do not intersect $\\{1,2,3\\}$ are the same in
$\mathbf{T}$ and $\mathbf{T}^{\prime}$.

  2. (2)

If $i=1$, then $c(3,\mathbf{T}^{\prime})=c(3,\mathbf{T})\cup\\{1,2\\}$.
Otherwise, $c(3,\mathbf{T}^{\prime})=c(3,\mathbf{T})\setminus\\{1,2\\}.$

  3. (3)

There is a c.s.p.b.
$\mu:\textup{OC}(\mathbf{T})\rightarrow\textup{OC}(\mathbf{T}^{\prime})$.

This follows by inspection. The next statement is a direct consequence of the
above and the definitions.

###### 4.5.2. COROLLARY.

Suppose $(\mathbf{T}_{1},\mathbf{T}_{2})\in
D_{\alpha\beta}(\mathcal{T}_{D}(M_{1},M_{2}))$ with
$\\{\alpha,\beta\\}=\\{\alpha_{1}^{\prime},\alpha_{3}\\}$. If
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})=T_{\alpha\beta}(\mathbf{T}_{1},\mathbf{T}_{2})$,
then there is a c.s.p.b
$\mu:\textup{OC}(\mathbf{T})\rightarrow\textup{OC}(\mathbf{T}^{\prime})$.

The next proposition states that, except for the obvious cases, the
$\tau$-invariant is unchanged by moving through cycles. It follows by an
argument identical to that of Proposition 3.1.4 for the roots $\alpha_{i}$
with $i\geq 2$ and direct inspection for $\alpha_{1}^{\prime}$.

###### 4.5.3. PROPOSITION.

Let $\mathbf{T}\in\mathcal{T}_{D}(n)$ and consider a cycle $c$ in $\mathbf{T}$
that is either open, or is a closed cycle with $c\neq\\{a,a+1\\}$ for some
$a\leq n-1$. Then $\tau(\mathbf{E}(\mathbf{T},c))=\tau(\mathbf{T})$.

In fact, unless $c$ is a closed cycle and $c=\\{1,2\\}$, if
$(\mathbf{T}_{1},\mathbf{T}_{2})$ is in the domain of $T_{\alpha\beta}$ for
$\\{\alpha,\beta\\}=\\{\alpha_{1}^{\prime},\alpha_{3}\\}$, then so is its
image under moving though $c.$ Even more is true. As in Proposition 3.1.5, the
two operations commute:

###### 4.5.4. PROPOSITION.

Suppose $(\mathbf{T}_{1},\mathbf{T}_{2})\in
D_{\alpha\beta}(\mathcal{T}_{D}(M_{1},M_{2}))$ with
$\\{\alpha,\beta\\}=\\{\alpha_{1}^{\prime},\alpha_{3}\\}$. Let
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})=T_{\alpha\beta}(\mathbf{T}_{1},\mathbf{T}_{2})$
and consider $c$, an extended cycle in $\mathbf{T}_{1}$ with respect to
$\mathbf{T}_{2}$ that either consists of open cycles or is closed and there is
a $k\in c$ with $k\not\in\\{1,2\\}$. In the former case let
$c^{\prime}=\mu(c)$, with $\mu$ as in Corollary 4.5.2, and in the latter, let
$c^{\prime}=ec(k,\mathbf{T}_{1}^{\prime};\mathbf{T}_{2}^{\prime})$. Then

$T_{\alpha\beta}(\mathbf{E}((\mathbf{T}_{1},\mathbf{T}_{2}),c,L))=\mathbf{E}((\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime}),c^{\prime},L)).$

###### Proof.

If $c\cap ec(3,\mathbf{T}_{1};\mathbf{T}_{2})=\varnothing$, then this result
is clear by Proposition 4.5.1 (1). So let
$c=ec(3,\mathbf{T}_{1};\mathbf{T}_{2})$. When $S_{22}\not\in
P(2,\mathbf{T}_{1})\cup P(3,\mathbf{T}_{1})$, the proof mirrors the argument
of the original Proposition 3.1.5. Otherwise, the result follows directly from
the definition of $T_{\alpha\beta}$, the subsequent remark, and Proposition
4.5.1. ∎

We now turn to the operators of the form $T_{\alpha\beta}$ where
$\\{\alpha,\beta\\}=\\{X,\gamma\\}$ for some $X\in\\{C,D\\}$ and
$\gamma\in\\{\alpha_{1}^{\prime},\alpha_{2},\alpha_{4}\\}$. We first describe
the effect of the replacement operation of Definition 4.4.8 on the cycle
structure of a tableau.

###### 4.5.5. PROPOSITION.

Suppose $\mathbf{T}\in\mathcal{T}_{D}(M)$, let $X$ and $Y$ be a replaceable
pair, and assume $X\subseteq\mathbf{T}$, writing
$\mathbf{T}^{\prime}=\textup{Re}(X,Y;\mathbf{T})$.

  1. (1)

If $k\in M$ but $k\notin\\{1,2,3,4\\}$, then
$P^{\prime}(k,\mathbf{T}^{\prime})=P^{\prime}(k,T)$ and in particular, cycles
that do not intersect the set $\\{1,2,3,4\\}$ are the same in $\mathbf{T}$ and
$\mathbf{T}^{\prime}$.

  2. (2)

The cycles through $1$ and $2$ coincide in both $\mathbf{T}$ and
$\mathbf{T}^{\prime}$. If $c=\\{1,2\\}$ is a closed cycle in $\mathbf{T}$,
then $c(3,\mathbf{T}^{\prime})=c(3,\mathbf{T})\cup\\{1,2\\}$. Otherwise $3\in
c(1,\mathbf{T})$.

  3. (3)

The cycle $c(4,\mathbf{T})$ is closed iff $3\in c(4,\mathbf{T}^{\prime})$. If
this is so, then $c(4,\mathbf{T}^{\prime})=c(3,\mathbf{T})\cup
c(4,\mathbf{T})$ or $c(3,\mathbf{T})\cup c(4,\mathbf{T})\setminus\\{1,2\\}$.

  4. (4)

If $c(4,\mathbf{T})$ is closed or $3\in c(4,\mathbf{T})$, then there is a
c.s.p.b
$\mu:\textup{OC}(\mathbf{T})\rightarrow\textup{OC}(\mathbf{T}^{\prime})$.

  5. (5)

If $c(4,\mathbf{T})$ is open and $3\notin c(4,\mathbf{T})$, then

$\textup{OC}(\mathbf{T})\setminus\bigcup_{i\leq
4}c(i,\mathbf{T})=\textup{OC}(\mathbf{T}^{\prime})\setminus\bigcup_{i\leq
4}c(i,\mathbf{T}^{\prime})$

and this equality gives a c.s.p.b. Further, we have
$S_{b}(c(4,\mathbf{T}))=S_{b}(c(4,\mathbf{T}^{\prime}))$ and
$S_{f}(c(4,\mathbf{T}))=S_{f}(c(3,\mathbf{T}^{\prime}))$, or
$S_{b}(c(4,\mathbf{T}))=S_{b}(c(3,\mathbf{T}^{\prime}))$ and
$S_{f}(c(4,\mathbf{T}))=S_{f}(c(4,\mathbf{T}^{\prime}))$.

This proposition follows via a detailed inspection. Combined with the
definitions, we obtain the following:

###### 4.5.6. COROLLARY.

Suppose that $(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})\in
T_{\alpha\beta}(\mathbf{T}_{1},\mathbf{T}_{2})$ where
$\\{\alpha,\beta\\}=\\{X,\gamma\\}$ for some $X\in\\{C,D\\}$,
$\gamma\in\\{\alpha_{1}^{\prime},\alpha_{2},\alpha_{4}\\}$, and
$(\mathbf{T}_{1},\mathbf{T}_{2})$ is of the appropriate left type. Then there
is a c.s.p.b.
$\mu:\textup{OC}(\mathbf{T})\rightarrow\textup{OC}(\mathbf{T}^{\prime})$.

Our next result shows that in some sense the replacement operation of
Definition 4.4.8 commutes with moving through the extended cycles that pass
through $3$ and $4$.

###### 4.5.7. PROPOSITION.

Consider $(\mathbf{T}_{1},\mathbf{T}_{2})\in\mathcal{T}_{D}(n,n)$. Let
$\overline{X}$ and $\overline{Y}$ be a replaceable pair and let
$X=\mathbf{E}(\overline{X},c(3,\overline{X}),c(4,\overline{X}))$ and
$Y=\mathbf{E}(\overline{Y},c(3,\overline{Y}),c(4,\overline{Y}))$. Then $X$ and
$Y$ form a replaceable pair. Let $c=ec(3,\mathbf{T}_{1};\mathbf{T}_{2}).$

  1. (1)

Suppose that $4\in c$. Let
$(\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2})=\mathbf{E}((\mathbf{T}_{1},\mathbf{T}_{2}),c,L)$
writing $c_{3}=ec(3,\overline{\mathbf{T}}_{1};\overline{\mathbf{T}}_{2})$ and
$c_{4}=ec(4,\overline{\mathbf{T}}_{1};\overline{\mathbf{T}}_{2})$. Then if
$\overline{X}\subseteq\overline{\mathbf{T}}_{1}$, we have
$X\subseteq\mathbf{T}_{1}$ and

$(\textup{Re}(\overline{X},\overline{Y};\overline{\mathbf{T}}_{1}),\overline{\mathbf{T}}_{2})=\mathbf{E}((\textup{Re}(X,Y;\mathbf{T}_{1}),\mathbf{T}_{2}),\\{c_{3},c_{4}\\},L).$

  2. (2)

Suppose that $4\notin c$ and let $d=ec(4,\mathbf{T}_{1};\mathbf{T}_{2})$. Let
$(\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2})=\mathbf{E}((\mathbf{T}_{1},\mathbf{T}_{2}),\\{c,d\\},L)$.
Then if $\overline{X}\subseteq\overline{\mathbf{T}}_{1}$, we have
$X\subseteq\mathbf{T}_{1}$ and

$(\textup{Re}(\overline{X},\overline{Y};\overline{\mathbf{T}}_{1}),\overline{\mathbf{T}}_{2})=\mathbf{E}((\textup{Re}(X,Y;\mathbf{T}_{1}),\mathbf{T}_{2}),c^{\prime},L).$

Finally, we would like to show that, in an appropriate sense, moving through
cycles commutes with the action of these operators. For this to make sense, we
first need to note:

###### 4.5.8. PROPOSITION.

Let $\mathbf{T}\in\mathcal{T}_{D}(n)$,
$\mathcal{X}\in\\{\mathcal{A}_{\beta},\mathcal{B}_{\beta},\mathcal{C},\mathcal{D}\\}$,
and consider an open cycle $c\in\textup{OC}(\mathbf{T})$ or a closed cycle
with $c\neq\\{1,2\\}$. Then $\mathbf{T}$ is of left type $\mathcal{X}$ iff
$\mathbf{E}(\mathbf{T},c)$ is of left type $\mathcal{X}$.

###### 4.5.9. PROPOSITION.

Consider an operator $T_{\alpha\beta}$ with
$\\{\alpha,\beta\\}=\\{X,\gamma\\}$ for some $X\in\\{C,D\\}$,
$\gamma\in\\{\alpha_{1}^{\prime},\alpha_{2},\alpha_{4}\\}$, and take
$(\mathbf{T}_{1},\mathbf{T}_{2})$ in its domain. Let $c$ be an extended cycle
in $\mathbf{T}_{1}$ relative to $\mathbf{T}_{2}$ and suppose that $c$ is not a
closed cycle with $c=\\{1,2\\}$.

  1. (1)

Suppose that $c\cap\\{3,4\\}=\varnothing$ so that
$c\cap\\{1,2,3,4\\}=\varnothing$, or that $c\cap\\{3,4\\}=\\{3,4\\}$. In the
former case, let $c^{\prime}=c$. In the latter, take
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})\in
T_{\alpha\beta}(\mathbf{T}_{1},\mathbf{T}_{2})$ and let
$c^{\prime}=ec(3,\mathbf{T}_{1}^{\prime};\mathbf{T}_{2}^{\prime})$. Then

$T_{\alpha\beta}(\mathbf{E}((\mathbf{T}_{1},\mathbf{T}_{2}),c,L))=\\{\mathbf{E}(Z,c^{\prime},L)\;|\;Z\in
T_{\alpha\beta}(\mathbf{T}_{1},\mathbf{T}_{2})\\}.$

  2. (2)

Suppose that $c\cap\\{3,4\\}$ is a singleton. Note that $T_{\alpha\beta}$ is
single-valued, write
$T_{\alpha\beta}((\mathbf{T}_{1},\mathbf{T}_{2}))=(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})$
and let
$c^{\prime}=ec(3,\mathbf{T}_{1}^{\prime};\mathbf{T}_{2}^{\prime})=ec(4,\mathbf{T}_{1}^{\prime};\mathbf{T}_{2}^{\prime}).$
Then

$T_{\alpha\beta}(\mathbf{E}((\mathbf{T}_{1},\mathbf{T}_{2}),c,L))=\mathbf{E}(T_{\alpha\beta}((\mathbf{T}_{1},\mathbf{T}_{2})),c^{\prime},L).$

###### Proof.

We first consider statement (2). It can be verified by considering the
individual cases that are used to define $T_{\alpha\beta}$. We relate one.
Suppose $c=ec(4,\mathbf{T}_{1},\mathbf{T}_{2})$ and the operator
$T_{\alpha\beta}$ is defined via the first case of Definition 4.4.10-(2). Let
$(\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2})=\mathbf{E}((\mathbf{T}_{1},\mathbf{T}_{2}),c,L)$
and note that
$T_{\alpha\beta}((\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2}))=(\textup{Re}(\overline{X},\overline{Y};\overline{\mathbf{T}}_{1}^{\prime}),\overline{\mathbf{T}}_{2}^{\prime})$
where
$(\overline{\mathbf{T}}_{1}^{\prime},\overline{\mathbf{T}}_{2}^{\prime})=\mathbf{E}((\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2}),ec(3,\overline{\mathbf{T}}_{1};\overline{\mathbf{T}}_{2}),L)$
and $\overline{X},\overline{Y}\in\mathcal{T}_{D}(4)$. Constructing $X$ and $Y$
as in Proposition 4.5.7, we have
$\mathbf{T}_{1}^{\prime}=\textup{Re}(X,Y;\mathbf{T}_{1})$. We would like to
know that

$(\textup{Re}(\overline{X},\overline{Y};\overline{\mathbf{T}}_{1}^{\prime}),\overline{\mathbf{T}}_{2}^{\prime})=\mathbf{E}((\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime}),c(3,\mathbf{T}_{1}^{\prime};\mathbf{T}_{2}^{\prime}),L),$

which follows from Proposition 4.5.7-(2).

The first part of statement (1) follows directly from Proposition 4.5.5-(1).
The second part follows an argument similar to that of statement (2) via a
case-by-case analysis and an application of Proposition 4.5.7. ∎

##

###### 4.6.1. DEFINITION.

We write $\Pi^{*}=\Pi^{\prime}\cup\\{A,B,C,D\\}$ and say that

$\Sigma=(\alpha^{1},\beta^{1}),\ldots,(\alpha^{k},\beta^{k})$

is a sequence for $\Pi^{*}$ if for each $i$ either
$\\{\alpha^{i},\beta^{i}\\}$ is a pair of adjacent simple roots in
$\Pi^{\prime}$ or $\\{\alpha^{i},\beta^{i}\\}=\\{X,\alpha\\}$ for some
$X\in\\{C,D\\}$ and
$\alpha\in\\{\alpha_{1}^{\prime},\alpha_{2},\alpha_{4}\\}$. Analogous to
Definition 3.2.1, we define the composition of operators $T_{\Sigma}^{L}$ and
$T_{\Sigma}^{R}$ for such a $\Sigma$. We also carry over to this situation the
analogues of any other notation introduced in Definition 3.2.1 and Definition
3.2.5.

###### REMARK.

Suppose $\Sigma$ is a sequence for $\Pi^{*}$ and
$(\mathbf{T}_{1},\mathbf{T}_{2})\in\mathcal{T}_{D}(M_{1},M_{2})$ with
$M_{1}=\\{1,\ldots,n\\}$. If
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})\in
T_{\Sigma}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$, then
$n_{v}((\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime}))\equiv
n_{v}((\mathbf{T}_{1},\mathbf{T}_{2}))\pmod{4}$. The purpose of this section
is to prove an analogue of Theorem 3.2.2.

###### 4.6.2. THEOREM.

Suppose $(\mathbf{T}_{1},\mathbf{T}_{2})$ and
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})\in\mathcal{T}_{D}(M_{1},M_{2})$
with $M_{1}=\\{1,\ldots,n\\}$ and suppose
$\mathbf{S}(\mathbf{T}_{2})=\mathbf{S}(\mathbf{T}_{2}^{\prime})$ and
$n_{v}((\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime}))\equiv
n_{v}((\mathbf{T}_{1},\mathbf{T}_{2}))\pmod{4}$. Then there is a sequence
$\Sigma$ for $\Pi^{*}$ such that
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})\in
T_{\Sigma}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$.

In broad outline, our proof of Theorem 4.6.2 will follow that of Theorem 3.2.2
as closely as possible. We will prove first Proposition 4.6.7 and Lemma 4.6.8,
the analogues of Proposition 3.2.4 and Lemma 3.2.9. The latter used as main
ingredients Lemmas 3.2.6, 3.2.7, and 3.2.8. The analogues of Lemmas 3.2.6 and
3.2.7 obtained by simply replacing $\mathcal{T}_{C}(M_{1},M_{2})$ by
$\mathcal{T}_{D}(M_{1},M_{2})$ and $\Pi$ by $\Pi^{*}$ are easily verified and
we will use them here citing them by their original numbers. On the other
hand, the analogue to Lemma 3.2.8, though valid, is useless because of the
parity condition
$n_{v}((\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime}))\equiv
n_{v}((\mathbf{T}_{1},\mathbf{T}_{2}))\pmod{4}$ in Theorem 4.6.2. We will
replace it with a combination of Propositions 4.6.4 and 4.6.5. This
combination is less powerful than the original, so as a result, the proofs of
Proposition 4.6.7 and Lemma 4.6.8 will have many more special cases than the
originals.

###### 4.6.3. DEFINITION.

We say a set $F\subseteq\mathcal{F}$ is proper if there is a tableau
$\mathbf{T}$ in some $\mathcal{T}(M)$ such that
$F=\textup{Shape}(\mathbf{T})$. We say that $F$ is of nested type if it is a
union of strictly nested hooks of even length, each set back from the previous
one by an odd number of squares. Note that any such shape is proper. For
example, the shape with partition $5221$ is of nested type, but $5311$ is not,
as the inner hook in the latter case is set back two squares from the outer
one. We will say $\mathbf{T}$ is of nested type if its underlying shape is of
nested type.

###### REMARK.

If $\mathbf{T}\in\mathcal{T}(M)$ and if $P$ is an extremal position in $T$,
then $\textup{Shape}(\mathbf{T})\setminus P$ is proper, since for example
$\textup{Shape}(\mathbf{T})\setminus P=\textup{Shape}(\mathbf{T}^{\prime})$,
where $(\mathbf{T}^{\prime},v,\epsilon)=\beta((\mathbf{T},P))$. If
Shape($\mathbf{T}$) is of nested type, then an easy induction (tiling the
hooks from the outside in) shows that this shape admits a unique tiling by
dominos. The next proposition shows shapes of nested type are the only ones
admitting a unique such tiling.

###### 4.6.4. PROPOSITION.

Let $\mathbf{T}\in\mathcal{T}_{D}(M)$ and suppose that Shape($\mathbf{T}$) is
not of nested type. Then there is a $\mathbf{T}^{\prime}\in\mathcal{T}_{D}(M)$
such that $\textup{Shape}(\mathbf{T}^{\prime})=\textup{Shape}(\mathbf{T})$ and
$n_{v}(\mathbf{T}^{\prime})\equiv n_{v}(\mathbf{T})+2\pmod{4}$.

###### Proof.

The proof is by induction on $|M|$. When $|M|=2$, the proposition is vacuous,
so assume $|M|>2$. Remove the largest domino from $\mathbf{T}$. If the
resulting tableau $\mathbf{T}^{\prime}$ does not have its shape of nested
type, then the proposition holds for $\mathbf{T}^{\prime}$, whence clearly it
holds for $\mathbf{T}$, so assume that $\mathbf{T}^{\prime}$ does have its
shape of nested type. Then the hypothesis guarantees that the largest domino
in $\mathbf{T}$ occupies a hook of length two of its own, set back an even
number of squares from the next innermost hook. Moreover, if the largest
domino in $\mathbf{T}$ is vertical, then the two squares to the left of it
must be occupied by a vertical domino; similarly, if the largest domino in
$\mathbf{T}$ is horizontal, then the two squares above it must be occupied by
a horizontal domino. Reversing the orientations of this largest domino and the
one to its left (if it is vertical) or above it (if it is horizontal) and
relabelling dominos appropriately, we get a tableau $\mathbf{T}^{\prime}$ with
the required property.

∎

###### REMARK.

The only special shapes of nested type in type $D$ are the single hooks with
first row of odd length and first column of even length. If a special shape
$S$ is not of nested type and $P$ is a boxed extremal position in it., then
$S\setminus P$ is either a single hook or again not of nested type.

###### 4.6.5. PROPOSITION.

Let $\mathbf{T}\in\mathcal{T}_{D}(M)$ and let $P$ be an extremal position in
$\mathbf{T}$. If $P$ is the largest domino in $\mathbf{T}$ or
$\textup{Shape}(\mathbf{T})\setminus P$ is not of nested type, there is a
$\mathbf{T}^{\prime}\in\mathcal{T}_{D}(M)$ such that $P(\sup
M,\mathbf{T}^{\prime})=P$,
$\textup{Shape}(\mathbf{T}^{\prime})=\textup{Shape}(\mathbf{T})$, and
$n_{v}(\mathbf{T}^{\prime})\equiv n_{v}(\mathbf{T})\pmod{4}$.

###### Proof.

This follows at once from the previous proposition and the remark after
Definition 4.6.3. ∎

###### REMARK.

If $\mathbf{T}$ is a tableau of nested type, then each of its dominos lies in
exactly one hook and each of its open cycles consists of the dominos lying in
some hook.

We next prove a special case of Theorem 4.6.2.

###### 4.6.6. LEMMA.

Suppose $(\mathbf{T}_{1},\mathbf{T}_{2})$ and
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})$ are pairs of domino
tableaux in $\mathcal{T}_{D}(M_{1},M_{2})$ with $M_{1}=\\{1,\ldots,n\\}$.
Suppose further that
$\mathbf{S}(\mathbf{T}_{2})=\mathbf{S}(\mathbf{T}_{2}^{\prime})$,
$n_{v}((\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime}))\equiv
n_{v}((\mathbf{T}_{1},\mathbf{T}_{2}))\pmod{4}$, and
$S_{2,2}\notin\textup{Shape}(\mathbf{T}_{1})$. Then there is a sequence
$\Sigma$ for $\Pi^{*}\setminus\\{\alpha_{1}^{\prime}\\}$ such that

$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})=T_{\Sigma}^{L}((\mathbf{T}_{1},\mathbf{T}_{2})).$

###### Proof.

We first note that our hypotheses in fact imply not only that
$\mathbf{S}(\mathbf{T}_{2})=\mathbf{S}(\mathbf{T}_{2}^{\prime})$, but in fact,
$\mathbf{T}_{2}^{\prime}=\mathbf{T}_{2}$ and that
$n_{v}((\mathbf{T}_{1},\mathbf{T}_{2}))=n_{v}((\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime}))$.
We proceed by induction on $n$. Note that when $n=1$ and $n=2$ the result is
trivial since in fact we must have
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})=(\mathbf{T}_{1},\mathbf{T}_{2})$.
So suppose that $n\geq 3$, letting $P=P(n,\mathbf{T}_{1})$ and
$P^{\prime}=P(n,\mathbf{T}_{1}^{\prime})$.

Assume first that $P=P^{\prime}$. Using Definition 3.2.5, we let

$(\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2})=(\mathbf{T}_{1},\mathbf{T}_{2})-L\text{
\hskip 14.45377pt and \hskip
14.45377pt}(\overline{\mathbf{T}}_{1}^{\prime},\overline{\mathbf{T}}_{2}^{\prime})=(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})-L.$

Note that $\overline{\mathbf{T}}_{2}^{\prime}=\overline{\mathbf{T}}_{2}$. By
induction, there is a sequence $\Sigma$ for
$\Pi^{*}\setminus\\{\alpha_{1}^{\prime},\alpha_{n}\\}$ such that
$(\overline{\mathbf{T}}_{1}^{\prime},\overline{\mathbf{T}}_{2}^{\prime})=T_{\Sigma}^{L}((\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2}))$.
Set
$(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})=T_{\Sigma}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$.
Since $\Sigma$ is a sequence for
$\Pi^{*}\setminus\\{\alpha_{1}^{\prime},\alpha_{n}\\}$, we have
$P(n,\mathbf{T}_{1}^{1})=P(n,\mathbf{T}_{1})$ and
$\mathbf{T}_{2}^{1}=\mathbf{T}_{2}$. Consequently
$(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})=(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})$
and the lemma holds in this case.

Now assume $P\neq P^{\prime}$. For simplicity we assume further that $P$ is
horizontal; the argument is analogous when $P$ is vertical. Hence
$P=\\{S_{1,r-1},S_{1,r}\\}$ where $r=\rho_{1}(\mathbf{T}_{1})$, and
$P^{\prime}=\\{S_{k-1,1},S_{k,1}\\}$, where $k=\kappa_{1}(\mathbf{T}_{1})$. If
$r\geq 5$ or if $r=4$ and $k=3$, set $P_{1}=\\{S_{1,r-3},S_{1,r-2}\\}$,
otherwise set $P_{1}=\\{S_{k-3,1},S_{k-2,1}\\}$. Let
$(\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2})=(\mathbf{T}_{1},\mathbf{T}_{2})-L$.
By inspection, there is a tableau
$\overline{\mathbf{T}}\in\mathcal{T}_{D}(n-1)$ such that
$\textup{Shape}(\overline{\mathbf{T}})=\textup{Shape}(\overline{\mathbf{T}}_{1})$,
$P(n-1,\overline{\mathbf{T}})=P^{\prime}$, and
$P(n-2,\overline{\mathbf{T}})=P_{1}$. It is also clear that
$n_{v}(\overline{\mathbf{T}})=n_{v}(\overline{\mathbf{T}}_{1})$. By induction
there is a sequence $\Sigma_{1}$ for
$\Pi^{*}\setminus\\{\alpha_{1}^{\prime},\alpha_{n}\\}$ such that
$(\overline{\mathbf{T}},\overline{\mathbf{T}}_{2})=T_{\Sigma}^{L}((\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2}))$.
Let
$(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})=T_{\Sigma}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$.
Then $\mathbf{T}_{2}^{1}=\mathbf{T}_{2}$. Let
$\Sigma_{2}=(\alpha_{n},\alpha_{n-1})$ and let
$(\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2})=T_{\Sigma_{2}}^{L}((\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})).$
Then $\mathbf{T}_{2}^{2}=\mathbf{T}_{2}^{1}$ and
$P(n,\mathbf{T}_{1}^{2})=P(n,\mathbf{T}_{1}^{\prime})$. By the case of the
lemma we have already proved, there is a sequence $\Sigma_{3}$ for
$\Pi^{*}\setminus\\{\alpha_{1}^{\prime},\alpha_{n}\\}$ such that
$T_{\Sigma_{3}}^{L}((\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2}))=(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})$.
Setting $\Sigma=\Sigma_{1}\Sigma_{2}\Sigma_{3}$ verifies the lemma in this
case.

∎

###### 4.6.7. PROPOSITION.

Let $(\mathbf{T}_{1},\mathbf{T}_{2})\in\mathcal{T}_{D}(M_{1},M_{2})$ with
$M_{1}=\\{1,\ldots,n\\}$ and suppose that
$S_{2,2}\in\textup{Shape}(\mathbf{T}_{1})$. Then there is a sequence $\Sigma$
for $\Pi^{*}$ and $(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})\in
T_{\Sigma}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$ such that
$\mathbf{T}_{2}^{\prime}$ is special.

###### Proof.

The proof uses induction on $n$. Note that the case $n=0$ is trivial and
assume by induction that both Theorem 4.6.2 and Proposition 4.6.7 are true
when $M_{1}=\\{1,\ldots,n-1\\}$. We first assume that the square $S_{2,2}\in
P(n,\mathbf{T}_{1})$. Let $r=\rho_{1}(\mathbf{T}_{1})$ and
$k=\kappa_{1}(\mathbf{T}_{1})$ and suppose that $\mathbf{T}_{2}$ is not
special. Then one of the following holds:

  1. (1)

$P(n,\mathbf{T}_{1})=\\{S_{2,1},S_{2,2}\\}$,

  2. (2)

$P(n,\mathbf{T}_{1})=\\{S_{2,2},S_{3,2}\\}$ and $r=2$,

  3. (3)

$P(n,\mathbf{T}_{1})=\\{S_{2,2},S_{3,2}\\}$ and $r>2$, or

  4. (4)

$P(n,\mathbf{T}_{1})=\\{S_{2,2},S_{2,3}\\}$ and $r$ is even.

We first specify $\Sigma$ in a few low-rank cases. If $n=3$ and we are in case
(1) or (2), then $\Sigma=(\alpha_{1}^{\prime},\alpha_{3})$ and
$(\alpha_{3},\alpha_{1}^{\prime})$ respectively satisfy the requirements of
the proposition. When $n=4$ and we are in case (3) or (4), then
$(\mathbf{T}_{1},\mathbf{T}_{2})\in
D_{\alpha,X}(\mathcal{T}_{D}(M_{1},M_{2}))$ for some $X\in\\{C,D\\}$ and every
$\alpha\in\\{\alpha_{1}^{\prime},\alpha_{2},\alpha_{4}\\}$. Either
$\Sigma=(\alpha_{1}^{\prime},X)$ or $(\alpha_{2},X)$ will satisfy the
requirement of the proposition.

Now assume that we are not in one of the low-rank cases treated above. The
sequence $\Sigma$ will be specified as a composition in several steps. We
begin by specifying domino positions $P_{1}$ and $P_{2}$. In case (1) and
$n\geq 4$, we note that $r\geq 6$ and set $P_{1}=\\{S_{1,r-1},S_{1,r}\\}$ and
$P_{2}=\\{S_{1,r-3},S_{1,r-2}\\}$. In case (2) we set
$P_{1}=\\{S_{k-1,1},S_{k,1}\\}$ and $P_{2}=\\{S_{k-3,1},S_{k-2,1}\\}$. In
cases (3) and (4) and $r\geq 5$ we let $P_{1}$ and $P_{2}$ be as in case (1),
otherwise let $P_{1}$ and $P_{2}$ be as in case (2).

Now let
$(\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2})=(\mathbf{T}_{1},\mathbf{T}_{2})-L$
and let $\overline{\mathbf{T}}_{1}^{1}\in\mathcal{T}_{D}(n-1)$ be such that
$\textup{Shape}(\overline{\mathbf{T}}_{1}^{1})=\textup{Shape}(\overline{\mathbf{T}}_{1})$,
$P(n-1,\overline{\mathbf{T}}_{1}^{1})=P_{1}$, and
$P(n-2,\overline{\mathbf{T}}_{1}^{1})=P_{2}$. It is clear that such a
$\overline{\mathbf{T}}_{1}^{1}$ exists and that
$n_{v}(\overline{\mathbf{T}}_{1}^{1})=n_{v}(\overline{\mathbf{T}}_{1})$. Set
$\overline{\mathbf{T}}_{2}^{1}=\overline{\mathbf{T}}_{2}$. But now
$S_{2,2}\notin\textup{Shape}(\overline{\mathbf{T}}_{1})$, so by Lemma 4.6.6,
there is a sequence $\Sigma_{1}$ for
$\Pi^{*}\setminus\\{\alpha_{1}^{\prime},\alpha_{n}\\}$ such that
$(\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1})=T_{\Sigma_{1}}^{L}((\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2})).$
Let
$(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})=\mathbf{T}_{\Sigma_{1}}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$.
Then let $\Sigma_{2}$ be such that $\Sigma_{2}=(\alpha,\beta)$ with
$\\{\alpha,\beta\\}=\\{\alpha_{n-1},\alpha_{n}\\}$ and
$T_{\Sigma_{2}}^{L}((\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1}))\neq\varnothing$,
and set
$(\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2})=T_{\Sigma_{2}}^{L}((\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1}))$.
Let
$(\overline{\mathbf{T}}_{1}^{2},\overline{\mathbf{T}}_{2}^{2})=(\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2})-L$.
Then $S_{2,2}\in\textup{Shape}(\overline{\mathbf{T}}_{1}^{2})$, so by
induction there is a sequence $\Sigma_{3}$ for
$\Pi^{*}\setminus\\{\alpha_{n}\\}$ and a
$(\overline{\mathbf{T}}_{1}^{\prime},\overline{\mathbf{T}}_{2}^{\prime})\in
T_{\Sigma_{3}}^{L}((\overline{\mathbf{T}}_{1}^{2},\overline{\mathbf{T}}_{2}^{2}))$
such that $\overline{\mathbf{T}}_{2}^{\prime}$ is special. Let
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})\in
T_{\Sigma_{3}}^{L}((\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2}))$ be such that
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})=(\overline{\mathbf{T}}_{1}^{\prime},\overline{\mathbf{T}}_{2}^{\prime})-L$.
Then, as is easily seen using Lemma 3.2.7-(1), we have that
$\mathbf{T}_{1}^{\prime}$ is special, so setting
$\Sigma=\Sigma_{1}\Sigma_{2}\Sigma_{3}$ verifies the conclusion of the
proposition.

Henceforth assume that $S_{2,2}\notin P(n,\mathbf{T}_{1})$. We begin as in the
proof of Proposition 3.2.4, that is, we let
$(\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2})=(\mathbf{T}_{1},\mathbf{T}_{2})-L$.
By induction, there is a sequence $\Sigma_{1}$ for
$\Pi^{*}\setminus\\{\alpha_{n}\\}$ and a
$(\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1})\in
T_{\Sigma_{1}}^{L}((\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2}))$
with $\overline{\mathbf{T}}_{2}^{1}$ special. By Lemma 3.2.6, there is a
$(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})\in
T_{\Sigma_{1}}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$ such that
$(\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1})=(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})-L$.
Now clearly either $\mathbf{T}_{2}^{1}$ is special, or
$P(n,\mathbf{T}_{1}^{1})$ is unboxed and $\\{n\\}$ forms and extended cycle in
$\mathbf{T}_{1}^{1}$ relative to $\mathbf{T}_{2}^{1}$. In the former case we
are done, so we assume the latter. Set $P=P(n,\mathbf{T}_{1}^{1})$. Since
$S_{2,2}$ is $\phi_{D}$-fixed, Lemma 3.2.7-(1) implies that $S_{2,2}\notin P.$

The rest of the proof will follow the general outline of Proposition 3.2.4,
although lacking Proposition 3.1.10, we will treat separately the situations
where $P$ is horizontal and $P$ is vertical. The proof proceeds by a case-by-
case analysis. For the most part, the cases we consider parallel those of the
proof Proposition 3.2.4 and are numbered accordingly.

In cases (1)–(6) and their transposed versions (1′)–(6′), we will define
domino positions $P_{1}$ and $P_{2}$. Letting
$S=\textup{Shape}(\overline{\mathbf{T}}_{1}^{1})$ and $S^{\prime}=S\setminus
P_{1}\setminus P_{2}$, we will see that either:

  1. $\bullet$

$P_{1}$ and $P_{2}$ are boxed and $S^{\prime}$ is not a single hook, implying
that $S^{\prime}$ is not nested by Remark 4.6.4, or that

  2. $\bullet$

$S^{\prime}$ is not nested by direct observation.

In this setting, Proposition 4.6.5 allows us to construct a tableau
$\overline{\mathbf{T}}_{1}^{2}\in\mathcal{T}_{D}(n-1)$ with
$\textup{Shape}(\overline{\mathbf{T}}_{1}^{2})=\textup{Shape}(\overline{\mathbf{T}}_{1}^{1})$,
$P(n-1,\overline{\mathbf{T}}_{1}^{2})=P_{1}$,
$P(n-2,\overline{\mathbf{T}}_{1}^{2})=P_{2}$, and
$n_{v}(\overline{\mathbf{T}}_{1}^{2})\equiv
n_{v}(\overline{\mathbf{T}}_{1}^{1})\pmod{4};$ the rest of the proof in these
cases will follow as in the proof of Proposition 3.2.4 interchanging the pairs
$(\alpha_{n-1},\alpha_{n})$ and $(\alpha_{n},\alpha_{n-1})$ when necessary.
Cases (3a) and (7) parallel case (7) of the proof of Proposition 3.2.4 and
will be treated directly. Finally, cases (3′a) and (6′a) will require a
different approach.

Assume first that $P$ is horizontal and set $P=\\{S_{ij},S_{i,j+1}\\}$. Then
$\phi_{D}(S_{ij})=Y$ and both $i$ and $j$ are odd. First assume $i\geq 3$ and
set $r=\rho_{i-2}(\mathbf{T}_{1}^{1})$ and $s=\rho_{i-1}(\mathbf{T}_{1}^{1})$.

Case 1. Here $r>s$. As in case (1) of the proof of Proposition 3.2.4, we have
$r\geq s+2$ and $s\geq j+2$. We set $P_{1}=\\{S_{i-2,r-1},S_{i-2,r}\\}$. If
$i\geq 5$, or if $i=3$ and $s>3$, we set $P_{2}=\\{S_{i-1,s-1},S_{i-1,s}\\}.$
If $i=3$ and $s=3$ and $r>5$, so in fact $r\geq 7$, we let
$P_{2}=\\{S_{i-2,r-3},S_{i-2,r-2}\\}$. Finally, if $i=3$, $s=3$, and $r=5$, we
set $P_{2}=\\{S_{1,3},S_{2,3}\\}$.

Case 2. Here $r=s>j+1$. We set $P_{1}=\\{S_{i-2,r},S_{i-1,r}\\}$. If $i\geq
5$, or if $i=3$ and $r>3$, we set $P_{2}=\\{S_{i-2,r-1},S_{i-1,r-1}\\}$. If
$i=3$ and $r=3$, then $n=4$. If $n_{v}(\mathbf{T}_{1}^{1})=3$, we define
$P_{2}$ as before, otherwise let $P_{2}=\\{S_{2,1},S_{2,2}\\}$.

Case 3. Here $r=s=j+1$ and we assume further that either $i\geq 5$ or $i=3$
and $j>1$. Let $P_{1}$ and $P_{2}$ be as in the first part of case (2).

Case 3a. Here $r=s=j+1$, $i=3$, and $j=1$ and we can specify the sequence
$\Sigma$ more directly. We have that either

$\mathbf{T}_{1}^{1}=$ $1$ $2$ $3$ or $\mathbf{T}_{1}^{1}=$ $1$ $2$ $3$ .

In the first case, we set $\Sigma_{2}=(\alpha_{1}^{\prime},\alpha_{3})$, in
the second
$\Sigma_{2}=(\alpha_{2},\alpha_{3}),(\alpha_{3},\alpha_{1}^{\prime})$. Then
$\Sigma=\Sigma_{1}\Sigma_{2}$ satisfies the requirements of the proposition.

In the next four cases we are still operating under the assumption that $P$ is
horizontal, but now $i=1$ and $r$ as well as $s$ will be redefined as needed.

Case 4. Here $S_{3,j-1}\in\textup{Shape}(\mathbf{T}_{1}^{1})$. Let
$r=\kappa_{j-1}(\mathbf{T}_{1}^{1}).$ Then because
$\overline{\mathbf{T}}_{1}^{1}$ is special, we know that $r$ is even, and in
particular, $r\geq 4$. If $j>3$, or if $j=3$ and $r\geq 6$, we let
$P_{1}=\\{S_{r-1,j-1},S_{r,j-1}\\}$ and $P_{2}=\\{S_{r-3,j-1},S_{r-2,j-1}\\}$.
If $j=3$ and $r=4$, we set $k=\kappa_{1}(\mathbf{T}_{1}^{1})$ and note that
$k$ is even since $\overline{\mathbf{T}}_{1}^{1}$ is special. If $k\geq 6$, we
set $P_{1}=\\{S_{k-1,1},S_{k,1}\\}$ and $P_{2}=\\{S_{3,2},S_{4,2}\\}$. If
$k=4$, we set $P_{1}=\\{S_{4,1},S_{4,2}\\}$ and $P_{2}=\\{S_{3,1},S_{3,2}\\}$.

Case 5. Here $S_{3,j-1}\notin\textup{Shape}(\mathbf{T}_{1}^{1})$ and
$S_{3,j-2}\in\textup{Shape}(\mathbf{T}_{1}^{1}).$ If $j>3$, so that $j\geq 5$,
we define $P_{1}$ and $P_{2}$ as in case (5) of the proof of Proposition
3.2.4. If $j=3$, we set $k=\kappa_{1}(\mathbf{T}_{1}^{1})$ and note that $k$
is again even. We set $P_{1}=\\{S_{k-1,1},S_{k,1}\\}$. If $k\geq 6$, we set
$P_{2}=\\{S_{k-3,1},S_{k-2,1}\\}$. If $k=4$, then $n=4$, and we have to
account for the number of horizontal dominos. If
$n_{h}(\overline{\mathbf{T}}_{1}^{1})=2$, we set
$P_{2}=\\{S_{2,1},S_{2,2}\\}$; if $n_{h}(\overline{\mathbf{T}}_{1}^{1})=0$, we
set $P_{2}=\\{S_{1,2},S_{2,2}\\}$.

Case 6. Here $S_{3,j-2}\notin\textup{Shape}(\mathbf{T}_{1}^{1})$ and $j>3$, so
that $j\geq 5$. We set $P_{1}=\\{S_{2,j-2},S_{2,j-1}\\}$ and
$P_{2}=\\{S_{1,j-2},S_{1,j-1}\\}$.

Case 7. Here $j=3$ and $S_{3,1}\notin\textup{Shape}(\mathbf{T}_{1}^{1})$ and
we can specify the sequence $\Sigma$ more directly. We have that either

$\mathbf{T}_{1}^{1}=$ $1$ $3$ $2$ or $\mathbf{T}_{1}^{1}=$ $1$ $2$ $3$ .

In the first case we set
$\Sigma_{2}=(\alpha_{3},\alpha_{2}),(\alpha_{1}^{\prime},\alpha_{3})$, in the
second we set $\Sigma_{2}=(\alpha_{3},\alpha_{1}^{\prime})$. Then
$\Sigma=\Sigma_{1}\Sigma_{2}$ satisfies the requirements of the proposition.

This exhausts the cases when $P$ is horizontal and henceforth we suppose that
$P$ is vertical, setting $P=\\{S_{ji},S_{j+1,i}\\}$. We note
$\phi_{D}(S_{ji})=Z$ which under our assumptions implies that both $i$ and $j$
are even. For the first three cases we will assume that $i\geq 4$ and set
$r=\kappa_{i-2}(\mathbf{T}_{1}^{1})$ and $s=\kappa_{i-1}(\mathbf{T}_{1}^{1})$.

Case 1′. Here $r>s$, and as in case (1), we have $r\geq s+2$ and $s\geq j+2$.
We set $P_{1}=\\{S_{r-1,i-2},S_{r,i-2}\\}$ and
$P_{2}=\\{S_{s-1,i-1},S_{s,i-1}\\}$.

Case 2′. Here $r=s>j+1$. We define the domino positions
$P_{1}=\\{S_{r,i-2},S_{r,i-1}\\}$ and $P_{2}=\\{S_{r-1,i-2},S_{r-1,i-1}\\}$.

Case 3′. Here $r=s=j+1$, and setting $k=\kappa_{1}(\mathbf{T}_{1}^{1})$ and
$t=\rho_{1}(\mathbf{T}_{1}^{1})$, we require further that either $i\geq 6$,
$j\geq 4$, $k>4$, or $t>5$. If $i\geq 6$ or $j\geq 4$, we let $P_{1}$ and
$P_{2}$ be as in case (2′). Now assume that $i=4$ and $j=2$. Since
$\overline{\mathbf{T}}_{1}^{1}$ is special, we know that $k$ is even and $t$
is odd. If $k>4$, we set $P_{2}=\\{S_{3,2},S_{3,3}\\}$ and
$P_{1}=\\{S_{k-1,1},S_{k,1}\\}$. If $k=4$ and $t>5$, we let
$P_{1}=\\{S_{t-1,1},S_{t,1}\\}$ and $P_{2}=\\{S_{t-3,1},S_{t-2,1}\\}$.

Case 3′a. Here $i=4$, $j=2$, $\kappa_{1}(\mathbf{T}_{1}^{1})=4$, and
$\rho_{1}(\mathbf{T}_{1}^{1})=5$. We cannot apply the standard argument to
this case as the hypotheses of Proposition 4.6.5 do not hold at one of the
steps, so we address it directly. Let $\overline{\mathbf{T}}$ be the tableau:

$\overline{\mathbf{T}}=$ $1$ $2$ $4$ $5$ $3$ $6$

If $n_{h}(\overline{\mathbf{T}}_{1}^{1})=4$, set
$(\overline{\mathbf{T}}_{1}^{2},\overline{\mathbf{T}}_{2}^{2})=(\overline{\mathbf{T}},\overline{\mathbf{T}}_{2}^{1})$.
If on the other hand we have $n_{h}(\overline{\mathbf{T}}_{1}^{1})=2$, set
$(\overline{\mathbf{T}}_{1}^{2},\overline{\mathbf{T}}_{2}^{2})=\mathbf{E}((\overline{\mathbf{T}},\overline{\mathbf{T}}_{2}^{1}),ec(1,\overline{\mathbf{T}};\overline{\mathbf{T}}_{2}^{1}),L).$
Then
$\mathbf{S}(\overline{\mathbf{T}}_{2}^{2})=\mathbf{S}(\overline{\mathbf{T}}_{2}^{1})$
and
$n_{v}((\overline{\mathbf{T}}_{1}^{2},\overline{\mathbf{T}}_{2}^{2}))=n_{v}((\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1}))$,
where we appeal to Proposition 4.2.4 for the latter if necessary. By induction
on Theorem 4.6.2, there is a sequence $\Sigma_{2}$ for
$\Pi^{*}\setminus\\{\alpha_{7}\\}$ such that
$(\overline{\mathbf{T}}_{1}^{2},\overline{\mathbf{T}}_{2}^{2})\in
T_{\Sigma_{2}}^{L}((\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1}))$.
Let $(\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2})\in
T_{\Sigma_{2}}^{L}((\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1}))$ be such that
$(\overline{\mathbf{T}}_{1}^{2},\overline{\mathbf{T}}_{2}^{2})=(\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2})-L$.
Let $\Sigma_{3}=(\alpha_{7},\alpha_{6})$ and define
$(\mathbf{T}_{1}^{3},\mathbf{T}_{2}^{3})=T_{\Sigma_{3}}^{L}((\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2}))$.
Further, let
$(\overline{\mathbf{T}}_{1}^{3},\overline{\mathbf{T}}_{2}^{3})=(\mathbf{T}_{1}^{3},\mathbf{T}_{2}^{3})-L$.
Then by induction on Proposition 4.6.7, there is a sequence $\Sigma_{4}$ for
$\Pi^{*}\setminus\\{\alpha_{7}\\}$ and a
$(\overline{\mathbf{T}}_{1}^{\prime},\overline{\mathbf{T}}_{2}^{\prime})\in
T_{\Sigma_{4}}^{L}((\overline{\mathbf{T}}_{1}^{3},\overline{\mathbf{T}}_{2}^{3}))$
such that $\overline{\mathbf{T}}_{2}^{\prime}$ is special. Let
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})\in
T_{\Sigma_{4}}^{L}((\mathbf{T}_{1}^{3},\mathbf{T}_{2}^{3}))$ be such that
$(\overline{\mathbf{T}}_{1}^{\prime},\overline{\mathbf{T}}_{2}^{\prime})=(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})-L$.
Using Lemma 3.2.7-(1) and the fact that
$S_{4,2}\notin\textup{Shape}(\mathbf{T}_{1}^{\prime})$, we have that
$P(7,\mathbf{T}_{1}^{\prime})$ is $\\{S_{3,2},S_{3,3}\\}$, so
$\mathbf{T}_{1}^{\prime}$ is special. That is,
$\Sigma=\Sigma_{1}\Sigma_{2}\Sigma_{3}\Sigma_{4}$ satisfies the requirement of
the proposition.

In the remaining cases we have $i=2$. Since $j$ must be even and we have
assumed that $S_{2,2}\notin P(n,\mathbf{T}_{1})$ and consequently
$S_{2,2}\notin P(n,\mathbf{T}_{1}^{1})$, we have that $j\geq 4$.

Case 4′. Here $S_{j-1,4}\in\textup{Shape}(\mathbf{T}_{1}^{1})$. Let
$t=\rho_{j-1}(\mathbf{T}_{1}^{1})$ and note that since
$\overline{\mathbf{T}}_{1}^{1}$ has special shape, $t$ must be odd and
consequently $t\geq 5$. We define the domino positions
$P_{1}=\\{S_{j-1,t-1},S_{j-1,t}\\}$ and $P_{2}=\\{S_{j-1,t-3},S_{j-1,t-2}\\}$.

Case 5′. Here $S_{j-1,4}\notin\textup{Shape}(\mathbf{T}_{1}^{1})$ but
$S_{j-2,4}\in\textup{Shape}(\mathbf{T}_{1}^{1})$. We proceed as in case (5),
but transposing everything.

Case 6′. Here $S_{j-2,4}\notin\textup{Shape}(\mathbf{T}_{1}^{1})$, and letting
$k=\kappa_{1}(\mathbf{T}_{1}^{1})$ and $t=\rho_{1}(\mathbf{T}_{1}^{1})$, we
further assume that either $j>4$, $j=4$ and $k>6$, or $j=4$, $k=6$, and $t\geq
5$. If $j>4$, we set $P_{1}=\\{S_{j-2,3},S_{j-1,3}\\}$ and
$P_{2}=\\{S_{j-2,2},S_{j-1,2}\\}$. If $j=4$ and $k>6$, we set
$P_{1}=\\{S_{k-1,1},S_{k,1}\\}$ and $P_{2}=\\{S_{k-3,1},S_{k-2,1}\\}$.
Finally, if $j=4$, $k=6$, and $t\geq 5$, we set
$P_{1}=\\{S_{1,t-1},S_{1,t}\\}$ and $P_{2}=\\{S_{3,2},S_{3,2}\\}$.

Case 6′a. We treat this case in the same fashion as case (3′a). We begin by
defining a domino tableau

$\overline{\mathbf{T}}=$ $1$ $3$ $5$ $6$ $2$ $4$

It is the image of the transpose of the tableau in case (3′a) under moving
through the cycle containing the domino with label 1. We can now follow the
steps of the proof of case (3′a) to construct the desired sequence $\Sigma$.

∎

###### 4.6.8. LEMMA.

Let $(\mathbf{T}_{1},\mathbf{T}_{2})$ be as in Theorem 4.6.2 and suppose that
$\mathbf{T}_{2}$ is special and $S_{2,2}\in\textup{Shape}(\mathbf{T}_{1}).$
Let $P^{\prime}$ be an extremal position in $\mathbf{T}_{1}$ and suppose that
there exists a tableau $\mathbf{T}\in\mathcal{T}_{D}(n)$ such that
$\textup{Shape}(\mathbf{T})=\textup{Shape}(\mathbf{T}_{1})$,
$P(n,\mathbf{T})=P^{\prime}$, and $n_{v}(\mathbf{T})\equiv
n_{v}(\mathbf{T}_{1})\pmod{4}$. Then there is a sequence $\Sigma$ for
$\Pi^{*}$ and a $(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})\in
T_{\Sigma}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$ such that
$P(n,\mathbf{T}_{1}^{\prime})=P^{\prime}$ and
$\mathbf{T}_{2}^{\prime}=\mathbf{T}_{2}$.

###### Proof.

As in the proof of Proposition 4.6.7, we will assume by induction on $n$ that
Theorem 4.6.2 is true when $M=\\{1,\ldots,n-1\\}$. Let
$P=P(n,\mathbf{T}_{1})$. We may assume that $P^{\prime}\neq P$. We first prove
the lemma under the additional assumption that both $P$ and $P^{\prime}$ are
boxed.

We will prove this portion of the lemma along the general lines of the
corresponding portion of Lemma 3.2.9. In place of Lemma 3.2.8 used in [5], we
will appeal to Propositions 4.6.4, 4.6.5, and the adjoining remarks. In what
follows, let
$(\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2})=(\mathbf{T}_{1},\mathbf{T}_{2})-L$.

A majority of the cases employ the same approach, which we call the standard
argument. In each such case we define a set $P_{1}$ which does not contain
$S_{2,2}$ and proceed as follows. If $P_{1}$ is boxed, then by Propositions
4.6.4 and 4.6.5, there is a tableau $\overline{\mathbf{T}}_{1}^{1}$ whose
shape matches that of $\overline{\mathbf{T}}_{1}$,
$P(n-1,\overline{\mathbf{T}}_{1}^{1})=P^{\prime}$,
$P(n-2,\overline{\mathbf{T}}_{1}^{1})=P_{1}$, and
$n_{v}(\overline{\mathbf{T}}_{1}^{1})\equiv
n_{v}(\overline{\mathbf{T}}_{1})\pmod{4}$. By induction on Theorem 4.6.2,
there is a sequence $\Sigma_{1}$ for $\Pi^{*}\setminus\\{\alpha_{n}\\}$ and
$(\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1})=T_{\Sigma_{1}}^{L}((\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2}))$
with $\overline{\mathbf{T}}_{2}^{1}=\overline{\mathbf{T}}_{2}$. Using Lemma
3.2.6, let $(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})\in
T_{\Sigma_{1}}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$ satisfying
$(\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1})=(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})-L$.
Lemma 3.2.7 now allows us to describe $P(n,\mathbf{T}_{1}^{1})$ and conclude
$\mathbf{T}_{2}^{1}=\mathbf{T}_{2}$. At this point, we can let
$\Sigma_{2}=(\alpha_{n-1},\alpha_{n})$ or $(\alpha_{n},\alpha_{n-1})$. Setting
$\Sigma=\Sigma_{1}\Sigma_{2}$ and
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})=T_{\Sigma_{2}}^{L}((\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{2}))$
completes the argument. If the suggested $P_{1}$ is not boxed, we let
$\widetilde{P}_{1}$ be the domino position sharing its bottom right square
with the bottom right square of $P_{1}$. In each case where this occurs,
$\widetilde{P}_{1}$ is boxed and the above argument can be made with
$\widetilde{P}_{1}$ in place of $P_{1}$. When the standard argument applies,
we will simply provide the appropriate $P_{1}$. Otherwise, we will provide
additional details.

In the first four cases, we assume that both $P$ and $P^{\prime}$ are
horizontal and set $P=\\{S_{i,j},S_{i,j+1}\\}$ and
$P^{\prime}=\\{S_{k,l},S_{k,l+1}\\}$. Since $P$ and $P^{\prime}$ are boxed, we
have that $j$ and $l$ are both even.

Case A. Here $k=i-1$. If $i>3$, $i=3$ and $l\geq 6$, or $i=2$ and $j\geq 4$,
let $P_{1}=\\{S_{i-1,l-2},S_{i-1,l-1}\\}$ and apply the standard argument. If
$i=2$ and $j=2$, we define $P_{1}$ as above, and modify the standard argument
by noting that the fact that the desired tableau
$\overline{\mathbf{T}}_{1}^{1}$ exists can be verified simply by inspection.
When $i=3$ and $l=4$ we examine several possibilities. For the remainder of
this case, let $r=\rho_{1}(\mathbf{T}_{1})$ and $s=\kappa_{1}(\mathbf{T}_{1})$
and note that since $\mathbf{T}_{1}$ is of special shape, $r$ is odd and $s$
is even.

  * (a)

Suppose $s\geq 6$. Let $P_{1}=\\{S_{s-1,1},S_{s,1}\\}$ and
$P_{2}=\\{S_{s-3,1},S_{s-2,1}\\}$. We begin as in the standard argument and
construct a sequence $\Sigma_{1}$ for $\Pi^{*}\setminus\\{\alpha_{n}\\}$ and
$(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})\in
T_{\Sigma_{1}}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$ with
$P(n-1,\mathbf{T}_{1}^{1})=P_{1}$, $P(n-2,\mathbf{T}_{1}^{1})=P_{2}$,
$P(n,\mathbf{T}_{1}^{1})=P$, and $\mathbf{T}_{2}^{1}=\mathbf{T}_{2}$. Let
$\Sigma_{2}=(\alpha_{n},\alpha_{n-1})$ and write
$(\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2})=T_{\Sigma_{2}}^{L}(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})$.
Following the same procedure, we can find a sequence $\Sigma_{3}$ for
$\Pi^{*}\setminus\\{\alpha_{n}\\}$ and
$(\mathbf{T}_{1}^{3},\mathbf{T}_{2}^{3})\in
T_{\Sigma_{3}}^{L}((\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2}))$ with
$P(n-1,\mathbf{T}_{1}^{3})=P^{\prime}$, $P(n-2,\mathbf{T}_{1}^{3})=P$,
$P(n,\mathbf{T}_{1}^{3})=P_{1}$ and $\mathbf{T}_{2}^{3}=\mathbf{T}_{2}$.
Letting $\Sigma_{4}=(\alpha_{n-1},\alpha_{n})$ and
$\Sigma=\Sigma_{1}\Sigma_{2}\Sigma_{3}\Sigma_{4}$, we obtain the desired
result.

  * (b)

Suppose $r\geq 7$ and let $P_{1}=\\{S_{1,r-1},S_{1,r}\\}$ and
$P_{2}=\\{S_{1,r-3},S_{1,r-2}\\}$. We argue as in (a) and first construct a
sequence $\Sigma_{1}$ for $\Pi^{*}\setminus\\{\alpha_{n}\\}$ and
$(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})\in
T_{\Sigma_{1}}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$ satisfying
$P(n,\mathbf{T}_{1}^{1})=P$, $P(n-2,\mathbf{T}_{1}^{1})=P^{\prime}$, and
$P(n-1,\mathbf{T}_{1}^{1})=P_{1}.$ Let $\Sigma_{2}=(\alpha_{n-1},\alpha_{n})$
and write
$(\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2})=T_{\Sigma_{2}}^{L}((\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1}))$.
Repeating the argument, we construct a sequence $\Sigma_{3}$ for
$\Pi^{*}\setminus\\{\alpha_{n}\\}$ and
$(\mathbf{T}_{1}^{3},\mathbf{T}_{2}^{3})\in
T_{\Sigma_{3}}^{L}((\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2}))$ with
$P(n,\mathbf{T}_{1}^{3})=P_{1}$, $P(n-2,\mathbf{T}_{1}^{3})=P_{2}$, and
$P(n-1,\mathbf{T}_{1}^{3})=P^{\prime}.$ Letting
$\Sigma_{4}=(\alpha_{n},\alpha_{n-1})$ and
$\Sigma=\Sigma_{1}\Sigma_{2}\Sigma_{3}\Sigma_{4}$, we obtain the desired
result.

  * (c)

Let $r=5$ and $s=4$. Define a tableau as follows:

$\overline{\mathbf{T}}=$ $1$ $3$ $4$ $5$ $6$ $2$

and let

$(\overline{\mathbf{T}}^{1}_{1},\overline{\mathbf{T}}^{1}_{2})=\begin{cases}(\overline{\mathbf{T}},\overline{\mathbf{T}}_{2})\hfill&\text{if
$n_{v}(\overline{\mathbf{T}}_{1})\equiv
n_{v}(\overline{\mathbf{T}})\pmod{4}$}\\\
\mathbf{E}((\overline{\mathbf{T}},\overline{\mathbf{T}}_{2}),ec(1,\overline{\mathbf{T}};\overline{\mathbf{T}}_{2}),L)\hfill&\text{
otherwise.}\\\ \end{cases}$

Then $\mathbf{S}(\overline{\mathbf{T}}_{2}^{1})=\overline{\mathbf{T}}_{2}$ and
$n_{v}(\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1})\equiv
n_{v}(\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2})\pmod{4}$ by
Proposition 4.2.4. Working inductively using Theorem 4.6.2, we can find a
sequence $\Sigma_{1}$ for $\Pi^{*}\setminus\\{\alpha_{n}\\}$ such that
$(\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1})\in
T_{\Sigma_{1}}^{L}(\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2}).$ Let
$(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})\in
T_{\Sigma_{1}}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$ be such that
$(\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1})=(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})-L$.
By Lemma 3.2.7, we know that $P(7,\mathbf{T}_{1}^{1})=P(7,\mathbf{T}_{1})$ or
$P^{\prime}(7,\mathbf{T}_{1})$. In the former case let
$\Sigma_{2}=(\alpha_{6},\alpha_{7})$, otherwise we define
$\Sigma_{2}=(\alpha_{6},\alpha_{7}),(\alpha_{4},\alpha_{5}),(\alpha_{4},C)$,
and write
$(\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2})=T_{\Sigma_{2}}^{L}((\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1}))$.
If $\mathbf{T}_{2}^{2}$ is special, then $P(7,\mathbf{T}_{1}^{2})=P^{\prime}$
and $\Sigma=\Sigma_{1}\Sigma_{2}$. If not, we can read off directly from
$\mathbf{T}_{1}^{2}$ that $P(7,\mathbf{T}_{1}^{2})$ is boxed and let
$(\overline{\mathbf{T}}_{1}^{2},\overline{\mathbf{T}}_{2}^{2})=(\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2})-L$.
Then by Proposition 4.6.7, there is a sequence $\Sigma_{3}$ for
$\Pi^{*}\setminus\\{\alpha_{7}\\}$ and
$(\overline{\mathbf{T}}_{1}^{3},\overline{\mathbf{T}}_{2}^{3})\in
T_{\Sigma_{3}}^{L}(\overline{\mathbf{T}}_{1}^{2},\overline{\mathbf{T}}_{2}^{2})$
with $\overline{\mathbf{T}}_{2}^{3}$ special. Let
$(\mathbf{T}_{1}^{3},\mathbf{T}_{2}^{3})\in
T_{\Sigma_{3}}^{L}(\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2})$ be such that
$(\mathbf{T}_{1}^{3},\mathbf{T}_{2}^{3})-L=(\overline{\mathbf{T}}_{1}^{3},\overline{\mathbf{T}}_{2}^{3})$.
Since $P(7,\mathbf{T}_{1}^{2})$ is boxed, Lemma 3.2.7-(4) implies
$P(7,\mathbf{T}_{1}^{3})=P^{\prime}$ and
$\Sigma=\Sigma_{1}\Sigma_{2}\Sigma_{3}$ gives the desired result.

Case B. Here $k<i-1$. Set $r=\rho_{i-1}(\mathbf{T}_{1})$ and note that $r\geq
j+1\geq 3$. If $i>3$ or $r>3$, then set $P_{1}=\\{S_{i-1,r-1},S_{i-1,r}\\}$.
Note that if $P_{1}$ is not boxed, we use $\widetilde{P}_{1}$ in place of
$P_{1}$, as defined above. If $i=3,$ $r=3$, and $l\geq 6$, let
$P_{1}=\\{S_{1,l-2},S_{1,l-1}\\}$. If $i=3$, $r=3$, and $l=4$, set
$P_{1}=\\{S_{1,3},S_{2,3}\\}$.

Case C. Here $k=i+1$. If $i\geq 3$, $i=2$ and $j\geq 6$, or $i=1$ and $l\geq
4$, let $P_{1}=\\{S_{i,j-2},S_{i,j-1}\\}.$ If $i=1$ and $l=2$, define $P_{1}$
in the same way and note that by one of the hypotheses of the lemma the
vertical dominos of $\mathbf{T}_{1}$ must all lie in its first column.

The final cases arise when $i=2$ and $j=4$. Let $r=\rho_{1}(\mathbf{T}_{1})$
and $s=\kappa_{1}(\mathbf{T}_{1})$. When $s\geq 6$ or $r\geq 7$, we mimic
parts (a) or (b) of case (A) accordingly. When $r=5$ and $s=4$, we proceed as
in part (c) of case (A), this time using the tableau

$\overline{\mathbf{T}}=$ $1$ $3$ $4$ $5$ $2$ $6$

and $\Sigma_{2}=(\alpha_{7},\alpha_{6})$ or
$\Sigma_{2}=(\alpha_{7},\alpha_{6}),(\alpha_{4},\alpha_{5}),(\alpha_{4},C)$.
If a $\mathbf{T}_{2}^{2}$ constructed in this way is special, then
$P(7,\mathbf{T}_{1}^{2})=P^{\prime}$ and we are done. Otherwise, we continue
as in part (c) defining $\Sigma_{3}$, this time noting that since
$S_{3,2}\not\in\overline{\mathbf{T}}_{1}^{3}$ as
$\overline{\mathbf{T}}_{2}^{3}$ is special, Lemma 3.2.7-(1) forces
$P(7,\mathbf{T}_{1}^{3})=P^{\prime}$.

Case D. Here $k>i+1$. Let $r=\rho_{k-1}(\mathbf{T}_{1})$ and note that $r\geq
l+1\geq 3$. If either $k\neq 3$ or $r\neq 3$, let
$P_{1}=\\{S_{k-1,r-1},S_{k-1,r}\\}$ and as in case (B) use $\widetilde{P}_{1}$
in place of $P_{1}$ when the latter is not boxed. If $k=3$, $r=3$, and $j>4$,
then let $P_{1}=\\{S_{1,j-2},S_{1,j-1}\\}$. If $k=3$, $r=3$, and $j=4$, we set
$P_{1}=\\{S_{1,3},S_{2,3}\\}$.

In the next six cases we assume that $P$ is horizontal and $P^{\prime}$ is
vertical, setting $P=\\{S_{ij},S_{i,j+1}\\}$ and
$P^{\prime}=\\{S_{kl},S_{k+1,l}\\}$. Since $P$ and $P^{\prime}$ are boxed, we
have that $j$ is even and $k$ is odd.

Case E. Here $k+1=i-1$. We set $P_{1}=\\{S_{i-2,l-1},S_{i-1,l-1}\\}$.

Case F. Here $k+1<i-1$. Let $r=\rho_{i-1}(\mathbf{T}_{1})$ and set
$P_{1}=\\{S_{i-1,r-1},S_{i-1,r}\\}$. As in case (B), use $\widetilde{P}_{1}$
in place of $P_{1}$ when the latter is not boxed.

Case G. Here $k=i+1$ and $l<j-1$. If $i\neq 2$ or $j\neq 4$, set
$P_{1}=\\{S_{i,j-2},S_{i,j-1}\\}$. When $i=2$ and $j=4$ we examine several
possibilities which parallel the subcases of case (A). In what follows, let
$r=\rho_{1}(\mathbf{T}_{1})$ and $s=\kappa_{1}(\mathbf{T}_{1})$ and note that
since $\mathbf{T}_{1}$ is of special shape, $r$ is odd and $s$ is even.

  1. (a)

If $r\geq 7$, let $P^{*}=\\{S_{1,r-1},S_{1,r}\\}$. We can first use case (A)
to move $n$ to $P^{*}$ and then employ the standard argument with $P_{1}=P$ to
move $n$ to $P^{\prime}$.

  2. (b)

If $s\geq 6$, we let $P_{1}=\\{S_{s-1,1},S_{s,1}\\}$ and
$P_{2}=\\{S_{s-3,1},S_{s-2,1}\\}$. Arguing as usual, we find a sequence
$\Sigma_{1}$ for $\Pi^{*}\setminus\\{\alpha_{n}\\}$ and
$(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})\in
T_{\Sigma_{1}}^{L}(\mathbf{T}_{1},\mathbf{T}_{2})$ with
$P(n-2,\mathbf{T}_{1}^{1})=P^{\prime}$, $P(n-1,\mathbf{T}_{1}^{1})=P_{1}$, and
$P(n,\mathbf{T}_{1}^{1})=P$. Let $\Sigma_{2}=(\alpha_{n},\alpha_{n-1})$ and
$(\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2})=T_{\Sigma_{2}}^{L}((\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1}))$.
Continuing, we can find a sequence $\Sigma_{3}$ for
$\Pi^{*}\setminus\\{\alpha_{n}\\}$ and
$(\mathbf{T}_{1}^{3},\mathbf{T}_{2}^{3})\in
T_{\Sigma_{3}}^{L}(\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2})$ with
$P(n-2,\mathbf{T}_{1}^{3})=P_{2}$, $P(n-1,\mathbf{T}_{1}^{3})=P^{\prime}$, and
$P(n,\mathbf{T}_{1}^{3})=P_{1}$. Letting
$\Sigma_{4}=(\alpha_{n-1},\alpha_{n})$ and
$\Sigma=\Sigma_{1}\Sigma_{2}\Sigma_{3}\Sigma_{4}$ completes the proof.

  3. (c)

If $r=5$ and $s=4$, define the tableau $\overline{\mathbf{T}}$ by

$\overline{\mathbf{T}}=$ $1$ $2$ $3$ $4$ $5$ or $\overline{\mathbf{T}}=$ $1$
$3$ $4$ $5$ $2$ $6$

if $l=1$ or $l=2$, respectively. In either case, if
$n_{v}(\overline{\mathbf{T}}_{1})\equiv n_{v}(\overline{\mathbf{T}})\pmod{4}$
we can use a version of the standard argument with
$\Sigma_{2}=(\alpha_{n},\alpha_{n-1})$. So suppose
$n_{v}(\overline{\mathbf{T}}_{1})\not\equiv
n_{v}(\overline{\mathbf{T}})\pmod{4}$ and let
$(\overline{\mathbf{T}}^{1}_{1},\overline{\mathbf{T}}^{1}_{2})=\mathbf{E}((\overline{\mathbf{T}},\overline{\mathbf{T}}_{2}),ec(1,\overline{\mathbf{T}};\overline{\mathbf{T}}_{2}),L)$.
We first examine $l=1$, noting that
$S_{2,3}\not\in\overline{\mathbf{T}}_{1}^{1}$. As usual, we can find a
sequence $\Sigma_{1}$ for $\Pi^{*}\setminus\\{\alpha_{n}\\}$ such that
$(\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1})\in
T_{\Sigma_{1}}^{L}(\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2}).$ Let
$(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})\in
T_{\Sigma_{1}}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$ be such that
$(\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1})=(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})-L$.
By Lemma 3.2.7, we know that $P(6,\mathbf{T}_{1}^{1})=\\{S_{2,3},S_{2,4}\\}$.
If $P(5,\mathbf{T}_{1}^{1})$ is vertical, let the sequence
$\Sigma_{2}=(\alpha_{6},\alpha_{5})$, otherwise let
$\Sigma_{2}=(\alpha_{6},\alpha_{5}),(\alpha_{3},\alpha_{4}),(\alpha_{1}^{\prime},\alpha_{3})$
and write
$(\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2})=T_{\Sigma_{2}}^{L}((\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1}))$.
In the latter case, $\mathbf{T}_{2}^{2}=\mathbf{T}_{2}$ and
$P(6,\mathbf{T}_{1}^{2})=P^{\prime}$, as desired. In the former,
$\mathbf{T}_{2}^{2}$ is not special and we continue as in part (c) of case
(A), again using Lemma 3.2.7-(4) to show $P(6,\mathbf{T}_{1}^{3})=P^{\prime}$.
When $l=2$, we argue as above but let
$\Sigma_{2}=(\alpha_{7},\alpha_{6}),(\alpha_{4},\alpha_{5}),(\alpha_{4},C)$,
noting that for one $(\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2})\in
T_{\Sigma_{2}}^{L}((\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1}))$ we have
$\mathbf{T}_{2}^{2}=\mathbf{T}_{2}$ and $P(7,\mathbf{T}_{1}^{2})=P^{\prime}$,
as desired.

Case H. Here $k>i+1$. Let $s=\rho_{i+1}(\mathbf{T}_{1})$. We first consider
the case $s\leq j-3$. If $i=2$ and $j=4$, then $l=1$ and $k\geq 5$ and we set
$P_{1}=\\{S_{k-2,1},S_{k-1,1}\\}$; otherwise we set
$P_{1}=\\{S_{i,j-2},S_{i,j-1}\\}$. If $s=j-1$, let
$r=\kappa_{j-1}(\textup{Shape}(\mathbf{T}_{1})\setminus P^{\prime})$ and set
$P_{1}=\\{S_{r-1,j-1},S_{r,j-1}\\}$. Finally, suppose $s=j-2$. Since
$\mathbf{T}_{1}$ is special and $P$ is boxed, we must have
$\phi_{D}(S_{ij})=Z$ and further
$r=\kappa_{j-2}(\textup{Shape}(\mathbf{T}_{1})\setminus P^{\prime})$ is even
and greater or equal to $i+2$. We can let $P_{1}=\\{S_{r-1,j-2},S_{r,j-2}\\}$.

Case I. Here $k=i-1$, implying $P\cap P^{\prime}=\\{S_{i,j+1}\\}$. Since both
$P$ and $P^{\prime}$ are boxed, we have that $\phi_{D}(S_{ij})=Z$ and that
both $i$ and $j$ are even. We consider several cases. After specifying domino
positions $P_{1}$ and $P_{2}$, or $P^{\prime\prime}$ the proof follows the
outline of the corresponding case of Lemma 3.2.9 unless otherwise indicated.
We note that when $P^{\prime\prime}$ is specified, it always satisfies the
hypothesis required of $P^{\prime}$ in the statement of this lemma allowing us
to adapt the argument from Lemma 3.2.9 to this setting.

  * (a)

Here $i\geq 4$. Let $r$ and $s$ be the lengths of the $i-2$ and $i-3$ rows of
$\mathbf{T}_{1}$ respectively, and assume first that $r=j+1$, and $i\geq 6$ or
$j\geq 4$. Set $P_{1}=\\{S_{i-2,j+1},S_{i-1,j+1}\\}$ and
$P_{2}=\\{S_{i-2,j},S_{i-1,j}\\}$. In both cases, $P_{1}$ and $P_{2}$ are not
boxed, so we can not make our usual appeal to the second part of Remark 4.6.4.
However, note that $\textup{Shape}(\mathbf{T}_{1})\setminus P\setminus
P_{1}\setminus P_{2}$ is special and consists of more than a single hook, so
by the first part of the above remark, it is not of nested type, as desired.

If $r=j+1$, $i=4$, $j=2$, and $\kappa_{1}(\mathbf{T}_{1})=4$, we set
$P_{1}=\\{S_{3,2},S_{3,3}\\}$ and $P_{2}=\\{S_{3,1},S_{4,1}\\}$. If $s=r>j+1$
or $s>r$, we define $P^{\prime\prime}$ as in Lemma 3.2.9 and continue as
therein.

If $r=j+1$, $i=4$, $j=2$, and $\kappa_{1}(\mathbf{T}_{1})\geq 6$, a more
elaborate argument is required. Let $P_{1}=\\{S_{2,3},S_{3,3}\\}$ and
$P_{2}=\\{S_{2,2},S_{3,2}\\}$. Using the usual notation, let
$\overline{\mathbf{T}}$ be a tableau of the same shape as
$\overline{\mathbf{T}}_{1}$ with $P(n-1,\overline{\mathbf{T}})=P_{1}$ and
$P(n-2,\overline{\mathbf{T}})=P_{2}$. If $n_{v}(\overline{\mathbf{T}})\equiv
n_{v}(\overline{\mathbf{T}}_{1})\pmod{4}$, then we can continue as above.
Otherwise let
$(\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1})=\mathbf{E}((\overline{\mathbf{T}},\overline{\mathbf{T}}_{2}),ec(1,\overline{\mathbf{T}};\overline{\mathbf{T}}_{2}),L)$.
Construct $\Sigma_{1}$ and $(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})$ as usual,
let $\Sigma_{2}=(\alpha_{n-1},\alpha_{n})$ and
$(\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2})=T_{\Sigma_{2}}^{L}((\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1}))$.
The tableau $\mathbf{T}_{2}^{2}$ is not special, so we argue as usual in this
situation and find $\Sigma_{3}$ and
$(\mathbf{T}_{1}^{3},\mathbf{T}_{2}^{3})\in
T_{\Sigma_{3}}^{L}((\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2}))$ with
$\mathbf{T}_{2}^{3}$ special and $P(n,\mathbf{T}_{1}^{3})=P^{\prime}$.

  * (b)

Here $j\geq 4$ and the proof is parallel to that of the corresponding case in
Lemma 3.2.9. We provide the essential details. Note that we might as well
assume that $i=2$. Let $r$ and $s$ be the lengths of the $j-1$ and $j-2$
columns of $\mathbf{T}_{1}$ respectively. If $r=2$, set
$P_{1}=\\{S_{1,j},S_{1,j+1}\\}$ and $P_{2}=\\{S_{1,j-1},S_{2,j-1}\\}$. If
$s=r>2$, let $P^{\prime\prime}=\\{S_{s,j-2},S_{s,j-1}\\}$ and argue as in the
analogous case of Proposition 3.2.9 but using a combination of the present
cases (C) or (D) and (E) or (F). If $s>r>2$, then let
$P^{\prime\prime}=\\{S_{s-1,j-2},S_{s,j-2}\\}$. Note that since
$\mathbf{T}_{1}$ is special, $s\geq r+2$, $\phi_{D}(S_{s,j-2})=Z$, and
$P^{\prime\prime}$ is boxed. We finish by arguing as above but this time using
cases (H) and (D′).

  * (c)

Here $i=2$ and $j=2$. If $t=\kappa_{1}(\mathbf{T}_{1})\geq 6$, let
$P^{\prime\prime}=\\{S_{t-1,1},S_{t,1}\\}$. We can use cases (H) and then (D′)
to move $n$ first to $P^{\prime\prime}$ and then to $P^{\prime}$. If $t=4$,
then $n=4$. One is tempted to use exactly the same idea, but instead of case
(H), the argument would have to use case (J) in whose proof we appeal to the
present case. Instead, note that $n_{v}(\mathbf{T}_{1})=2$, and arguing as
usual, we can find a sequence $\Sigma_{1}$ such that
$(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})\in
T_{\Sigma_{1}}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$ where

$\mathbf{T}_{1}^{1}=$ $1$ $2$ $4$ $3$

Applying $\Sigma_{2}=(\alpha_{2},D)$ achieves the desired result. If $t=2$,
then $n=3$ and it is sufficient to let $\Sigma=(\alpha_{2},\alpha_{3})$.

Case J. Here $k=i+1$ and $l=j-1$ and the proof is identical to the
corresponding case of Lemma 3.2.9.

In the next four cases we assume that both $P$ and $P^{\prime}$ are vertical
and set $P=\\{S_{ji},S_{j+1,i}\\}$ and $P^{\prime}=\\{S_{l,k},S_{l+1,k}\\}$.
Since $P$ and $P^{\prime}$ are boxed, we know that both $j$ and $l$ are odd.

Case A′. Here $k=i-1$. If $i\neq 3$ or $l\neq 3$, let
$P_{1}=\\{S_{l-2,k},S_{l-1,k}\\}$. If $i=3$ and $l=3$, again set
$P_{1}=\\{S_{l-2,k},S_{l-1,k}\\}$, define
$(\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2})=(\mathbf{T}_{1},\mathbf{T}_{2})-L$,
and let $\overline{\mathbf{T}}$ be a tableau of the same shape as
$\overline{\mathbf{T}}_{1}$ but with
$P(n-1,\overline{\mathbf{T}}_{1})=P^{\prime}$ and
$P(n-2,\overline{\mathbf{T}}_{1})=P_{1}$. If
$n_{v}(\overline{\mathbf{T}})\equiv n_{v}(\overline{\mathbf{T}}_{1})\pmod{4}$,
the standard argument applies. If $n_{v}(\overline{\mathbf{T}})\not\equiv
n_{v}(\overline{\mathbf{T}}_{1})\pmod{4}$, there are two cases.

  1. (a)

If $\kappa_{1}(\mathbf{T}_{1})\geq 6$, let
$(\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1})=\mathbf{E}((\overline{\mathbf{T}},\overline{\mathbf{T}}_{2},ec(1,\overline{\mathbf{T}};\overline{\mathbf{T}}_{2}),L)$.
Let $\Sigma_{1}$ be a sequence for $\Pi^{*}\setminus\\{\alpha_{n}\\}$ so that
$(\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1})\in
T_{\Sigma_{1}}^{L}((\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2}))$ and
let $(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})\in
T_{\Sigma_{1}}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$ satisfy
$(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})-L=(\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1})$.
Then $P(n,\mathbf{T}_{1}^{1})=P(n,\mathbf{T}_{1})$ or
$P^{\prime}(n,\mathbf{T}_{1})$. Let $\Sigma_{2}=(\alpha_{n},\alpha_{n-1})$ and
define
$(\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2})=T_{\Sigma_{2}}^{L}((\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1}))$.
Since $\mathbf{T}_{2}^{2}$ is not special, we argue by induction. Consider
$(\overline{\mathbf{T}}_{1}^{2},\overline{\mathbf{T}}_{2}^{2})=(\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2})-L$
and let $\Sigma_{3}$ be a sequence for $\Pi^{*}\setminus\\{\alpha_{n}\\}$ such
that $(\overline{\mathbf{T}}_{1}^{3},\overline{\mathbf{T}}_{2}^{3})\in
T_{\Sigma_{3}}^{L}((\overline{\mathbf{T}}_{1}^{2},\overline{\mathbf{T}}_{2}^{2}))$
is special. Let $(\mathbf{T}_{1}^{3},\mathbf{T}_{2}^{3})\in
T_{\Sigma_{3}}^{L}((\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2}))$ satisfy
$(\mathbf{T}_{1}^{3},\mathbf{T}_{2}^{3})-L=(\overline{\mathbf{T}}_{1}^{3},\overline{\mathbf{T}}_{2}^{3})$.
Then $P(n,\mathbf{T}_{1}^{3})=P^{\prime}$ and
$\mathbf{T}_{2}^{3}=\mathbf{T}_{2}$.

  2. (b)

If $\kappa_{1}(\mathbf{T}_{1})=4$, arguing as usual we can find a sequence
$\Sigma_{1}$ such that $(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})\in
T_{\Sigma_{1}}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$ where

$\mathbf{T}_{1}^{1}=$ $1$ $2$ $5$ $3$ $4$

After applying $\Sigma_{2}=(\alpha_{5},\alpha_{4})$ we are in case (I) above.
Letting $\Sigma_{3}$ be the sequence of operators constructed therein, we
define $\Sigma=\Sigma_{1}\Sigma_{2}\Sigma_{3}$.

Case B′. Here $k<i-1$. Let $r=\kappa_{i-1}(\mathbf{T}_{1})$. If $i=3$, $r=2$,
and $l=3$, then $\Sigma$ is easy to find by inspection. If $i=3$, $r=2$, and
$l\geq 5$, let $P_{1}=\\{S_{l-2,1},S_{l-1,1}\\}$. Otherwise, note that if $r$
is odd, then since $\mathbf{T}_{1}$ is special, we have
$\kappa_{i-2}(\mathbf{T}_{1})=r$, $k\leq i-3$, and we can let
$P_{1}=\\{S_{r,i-2},S_{r,i-1}\\}$. If $r$ is even, then let
$P_{1}=\\{S_{r-1,i-1},S_{r,i-1}\\}$.

Case C′. Here $k=i+1$. If either $i\geq 3$, $i=2$ and $j\geq 5$, or $i=1$ and
$l\geq 3$, let $P_{1}=\\{S_{j-2,i},S_{j-1,i}\\}$. If $i=1$ and $l=1$, then the
hypotheses of the lemma allow us to consider just those tableaux
$\mathbf{T}_{1}$ all of whose dominos are vertical; we can define $P_{1}$ in
the same way. If $i=2$ and $j=3$, we consider two cases:

  1. (a)

If $t=\kappa_{1}(\mathbf{T}_{1})\geq 6$, let
$P^{\prime\prime}=\\{S_{t-1,1},S_{t,1}\\}$. Using case (A′) we can first move
$n$ to $P^{\prime\prime}$ and then to $P^{\prime}$ using case (D′).

  2. (b)

If $\kappa_{1}(\mathbf{T}_{1})=4$ and $n_{v}(\mathbf{T}_{1})=5$, we can let
$P_{1}=\\{S_{1,2},S_{2,2}\\}$. Otherwise, arguing as usual we can find a
sequence $\Sigma_{1}$ such that $(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})\in
T_{\Sigma_{1}}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$ where

$\mathbf{T}_{1}^{1}=$ $1$ $4$ $2$ $3$ $5$

Applying $\Sigma_{2}=(\alpha_{1}^{\prime},\alpha_{3}),(\alpha_{4},\alpha_{5})$
completes the argument.

Case D′. Here $k>i+1$. Let $r=\kappa_{k-1}(\mathbf{T}_{1})$. If $k=3$, $r=2$,
and $j=3$, then $n=4$ and either $\Sigma=(\alpha_{3},\alpha_{4})$ or
$\Sigma=(\alpha_{2},\alpha_{3}),(\alpha_{3},\alpha_{4})$ satisfies our needs.
If $k=3$, $r=2$, and $j\geq 5$, set $P_{1}=\\{S_{j-2,1},S_{j-1,1}\\}$.
Otherwise, note that if $r$ is odd, then since $\mathbf{T}_{1}$ is special, we
have $\kappa_{k-2}(\mathbf{T}_{1})=r$, $i\leq k-3$, and we can let
$P_{1}=\\{S_{r,k-2},S_{r,k-1}\\}$. If $r$ is even, then let
$P_{1}=\\{S_{r-1,k-1},S_{r,k-1}\\}$.

In the next six cases we assume that $P$ is vertical and $P^{\prime}$ is
horizonal, setting $P=\\{S_{j,i},S_{j+1,i}\\}$ and
$P^{\prime}=\\{S_{l,k},S_{l,k+1}\\}$. Since both are boxed, we have that $j$
is odd and $k$ is even.

Case E′. Here $k+1=i-1$. We let $P_{1}=\\{S_{l-2,i-1},S_{l-1,i-1}\\}$ and we
note that if this domino position is not boxed, we replace it with
$\widetilde{P_{1}}$ instead.

Case F′. Here $k+1<i-1$. We let $r=\kappa_{i-1}(\mathbf{T}_{1})$. If $r$ is
odd, then since $\mathbf{T}_{1}$ is special, we have
$\kappa_{i-2}(\mathbf{T}_{1})=r$, $k\leq i-4$, and we can let
$P_{1}=\\{S_{r,i-2},S_{r,i-1}\\}$. If $r$ is even, then let
$P_{1}=\\{S_{r-1,i-1},S_{r,i-1}\\}$.

Case G′. Here $k=i+1$ and $l<j-1$. Note first that $i$ is odd. If $i\geq 3$ or
$i=1$ and $l>2$, let $P_{1}=\\{S_{j-2,i},S_{j-1,i}\\}$. The case $i=1$ and
$l=1$ is excluded by the hypotheses of this lemma since
$S_{22}\notin\textup{Shape}(\mathbf{T}_{1})$. If $i=1$ and $l=2$, note that by
a hypothesis of the lemma, we need only to consider those $\mathbf{T}_{1}$
where $S_{22}$ is contained in a horizontal domino. Define $P_{1}$ as above.
Arguing as usual we can find a sequence $\Sigma_{1}$ such that
$(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})\in
T_{\Sigma_{1}}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$ with
$P(n-2,\mathbf{T}_{1}^{1})=P_{1}$, $P(n-1,\mathbf{T}_{1}^{1})=P^{\prime}$, and
$P(n,\mathbf{T}_{1}^{1})=P$. Applying $\Sigma_{2}=(\alpha_{n-1},\alpha_{n})$
completes the argument.

Case H′. Here $k>i+1$. Let $t=\kappa_{i+1}(\mathbf{T}_{1})$. We consider three
possibilities.

  * (a)

Here $t\leq j-3$. Let $P_{1}=\\{S_{j-2,i},S_{j-1,i}\\}$.

  * (b)

Here $t=j-1$. Let $r=\rho_{j-1}(\textup{Shape}(\mathbf{T}_{1})\setminus
P^{\prime})$. If $j\geq 5$, or if $j=3$ and $r\geq 4$, let
$P_{1}=\\{S_{j-1,r-1},S_{j-1,r}\\}$ and note that in certain circumstances it
may be necessary to replace $P_{1}$ by $\widetilde{P}_{1}$. The case $j=3$ and
$r=1$ is straightforward, so consider $j=3$ and $r=3$. If $l=1$, let
$P_{1}=\\{S_{1,k-2},S_{1,k-1}\\}$ if $k>4$ and $P_{1}=\\{S_{1,3},S_{2,3}\\}$
if $k=4$. If $l=2$ and $s=\rho_{1}(\mathbf{T}_{1})\geq 7$, let
$P^{\prime\prime}=\\{S_{1,s-1},S_{1,s}\\}$. By using the $l=1$ subcase above,
we can first move $n$ to $P^{\prime\prime}$ and then use case (C) to move it
to $P^{\prime}$. If $l=2$ and $u=\kappa_{1}(\mathbf{T}_{1})\geq 6$, let
$P^{\prime\prime}=\\{S_{u-1,1},S_{u,1}\\}$. Using case (A′) we can move $n$ to
$P^{\prime\prime}$ and use the $j\geq 5$ subcase above to move it to
$P^{\prime}$. Finally, assume $l=2$, $u=4$, and $s=5$. Let

$\overline{\mathbf{T}}=$ $1$ $2$ $3$ $4$ $5$ or $\overline{\mathbf{T}}=$ $1$
$3$ $4$ $5$ $6$ $2$

if $i=1$ or $i=2$, respectively. In either case, if
$n_{v}(\overline{\mathbf{T}}_{1})\equiv n_{v}(\overline{\mathbf{T}})\pmod{4}$
we can argue along the lines of the standard argument with
$\Sigma_{2}=(\alpha_{n-1},\alpha_{n})$. So suppose
$n_{v}(\overline{\mathbf{T}}_{1})\not\equiv
n_{v}(\overline{\mathbf{T}})\pmod{4}$ and let
$(\overline{\mathbf{T}}^{1}_{1},\overline{\mathbf{T}}^{1}_{2})=\mathbf{E}((\overline{\mathbf{T}},\overline{\mathbf{T}}_{2}),ec(1,\overline{\mathbf{T}};\overline{\mathbf{T}}_{2}),L)$.
As usual, we can find a sequence $\Sigma_{1}$ for
$\Pi^{*}\setminus\\{\alpha_{n}\\}$ such that
$(\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1})\in
T_{\Sigma_{1}}^{L}(\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2}).$ Let
$(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})\in
T_{\Sigma_{1}}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$ be such that
$(\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1})=(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})-L$.
Define $\Sigma_{2}=(\alpha_{n-1},\alpha_{n})$ and
$(\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2})=T_{\Sigma_{2}}^{L}((\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1}))$.
Then $\mathbf{T}_{2}^{2}$ is not special. Let
$(\overline{\mathbf{T}}_{1}^{2},\overline{\mathbf{T}}_{2}^{2})=(\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2})-L$.
Using induction, let $\Sigma_{3}$ be a sequence for
$\Pi^{*}\setminus\\{\alpha_{n}\\}$ so that
$(\overline{\mathbf{T}}_{1}^{3},\overline{\mathbf{T}}_{2}^{3})\in
T_{\Sigma_{3}}^{L}((\overline{\mathbf{T}}_{1}^{2},\overline{\mathbf{T}}_{2}^{2}))$
is special. If $(\mathbf{T}_{1}^{3},\mathbf{T}_{2}^{3})\in
T_{\Sigma_{3}}^{L}((\mathbf{T}_{1}^{2},\mathbf{T}_{2}^{2}))$ satisfies
$(\mathbf{T}_{1}^{3},\mathbf{T}_{2}^{3})-L=(\overline{\mathbf{T}}_{1}^{3},\overline{\mathbf{T}}_{2}^{3})$,
then $P(n,\mathbf{T}_{1}^{3})=P^{\prime}$ and
$\mathbf{T}_{2}^{3}=\mathbf{T}_{2}$.

  * (c)

Here $t=j-2$. Note that $\phi_{D}(S_{ji})=Y$ since $\mathbf{T}_{1}$ is special
and $P^{\prime}$ is boxed. Because $P^{\prime}$ is horizontal,
$r=\rho_{j-2}(\textup{Shape}(\mathbf{T}_{1})\setminus P^{\prime})\geq i+2$ and
must odd. We set $P_{1}=\\{S_{j-2,r-1},S_{j-2,r}\\}$.

Cases I′. Here $k=i-1$. This case parallels case (I), albeit with fewer
special subcases. We omit the arguments.

Cases J′. Here $k=i+1$ and $l=j-1$ and note that since $P$ and $P^{\prime}$
are boxed, $j\geq 3$. When $i\geq 3$, the proof parallels that of case (J).
When $i=1$, let $r=\rho_{l-1}(\mathbf{T}_{1})$. If $r=3$, we let
$P^{\prime\prime}=\\{S_{l-1,3},S_{l,3}\\}$. If $r>3$, then $r\geq 5$ and we
let $P^{\prime\prime}=\\{S_{l-1,r-1},S_{l-1,r}\\}$. In both cases
$P^{\prime\prime}$ is boxed and we argue by moving $n$ first to
$P^{\prime\prime}$ and then to $P^{\prime}$ using the previous cases.

This exhausts all of the possible cases, concluding the proof of the lemma
when both $P$ and $P^{\prime}$ are boxed.

We now assume that $P^{\prime}$ is boxed and $P$ is unboxed. We will closely
follow the proof of Lemma 3.2.9. First assume that $P$ is horizontal, setting
$P=\\{S_{ij},S_{i,j+1}\\}$. Under these hypotheses, we have
$\phi_{D}(S_{ij})=W$ or $Y$, but since $\mathbf{T}_{1}$ is special, it follows
that in fact $\phi_{D}(S_{ij})=W$ and $i\geq 2$. Furthermore,
$\rho_{i-1}(\mathbf{T}_{1})=j+1$ for otherwise $S_{i,j+2}$ would form an empty
hole. Set
$(\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2})=(\mathbf{T}_{1},\mathbf{T}_{2})-L$.
By Lemma 4.6.7 or by inspection, there is a sequence $\Sigma_{1}$ for
$\Pi^{*}\setminus\\{\alpha_{n}\\}$ and a
$(\overline{\mathbf{T}}^{1}_{1},\overline{\mathbf{T}}^{1}_{2})\in
T_{\Sigma_{1}}^{L}((\overline{\mathbf{T}}_{1},\overline{\mathbf{T}}_{2}))$
such that $\overline{\mathbf{T}}_{2}^{1}$ is special. Let
$(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})\in
T_{\Sigma_{1}}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$ be such that
$(\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1})=(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})-L$.
Then
$\textup{Shape}(\overline{\mathbf{T}}_{1}^{1})=(\textup{Shape}(\overline{\mathbf{T}}_{1})\setminus\\{S_{i-1,j+1}\\})\cup\\{S_{ij}\\}$,
and from Lemma 3.2.7-(1), we can conclude that

$P(n,\mathbf{T}_{1}^{1})=P^{\prime}(n,\mathbf{T}_{1})=\\{S_{i-1,j+1},S_{i,j+1}\\}.$

This domino position is boxed, and the therefore the cases of our lemma that
we have already proved apply. Hence there is a sequence $\Sigma_{2}$ for
$\Pi^{*}$ and a $(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})\in
T_{\Sigma_{2}}^{L}((\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1}))$ such that
$P(n,\mathbf{T}_{1}^{\prime})=P^{\prime}$. We set
$\Sigma=\Sigma_{1}\Sigma_{2}$. When $P=\\{S_{ij},S_{i+1,j}\\}$ is vertical
instead, we can conclude that $\phi_{D}(S_{ij})=W$, $j\geq 3$, and
$\kappa_{j-1}(\mathbf{T}_{1})=i+1$. A transposed version of the argument above
applies.

Finally, consider the case when $P^{\prime}$ is unboxed. First assume that
$P^{\prime}$ is horizontal, setting $P^{\prime}=\\{S_{ij},S_{i,j+1}\\}$.
Arguing as above, we find $\phi_{D}(S_{ij})=W$ and
$\rho_{i-1}(\mathbf{T}_{1})=j+1$. Set
$P^{\prime\prime}=\\{S_{i-1,j+1},S_{i,j+1}\\}$. It is boxed and an extremal
position in $\mathbf{T}_{1}$. Using Propositions 4.6.5 and 4.6.4 if necessary,
we can appeal to the cases of the lemma that we have already verified, letting
$\Sigma_{1}$ be a sequence for $\Pi^{*}$ satisfying
$(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})\in
T_{\Sigma_{1}}^{L}((\mathbf{T}_{1},\mathbf{T}_{2}))$ with
$\mathbf{T}_{2}^{1}=\mathbf{T}_{2}$ and
$P(n,\mathbf{T}_{1}^{1})=P^{\prime\prime}$. Now let
$(\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1})=(\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1})-L$.
By Theorem 2.2.3, there is an open cycle
$c\in\textup{OC}(\overline{\mathbf{T}}_{2}^{1})$ with $S_{b}(c)=S_{ij}$ and
$S_{f}(c)=S_{i-1,j+1}$. Again using Propositions 4.6.5 and 4.6.4 if necessary
we can find a tableau $\overline{\mathbf{T}}$ such that
$\textup{Shape}(\overline{\mathbf{T}})=\textup{Shape}(\overline{\mathbf{T}}_{1}^{1})$,
$P(n-1,\overline{\mathbf{T}})=\\{S_{i-1,j},S_{ij}\\}$, and
$n_{v}(\overline{\mathbf{T}})\equiv
n_{v}(\overline{\mathbf{T}}_{1}^{1})+2\pmod{4}$. Then $\\{n-1\\}$ constitutes
an extended cycle in $\overline{\mathbf{T}}$ relative to
$\overline{\mathbf{T}}_{2}^{1}$. Set

$(\overline{\mathbf{T}}_{1}^{2},\overline{\mathbf{T}}_{2}^{2})=\mathbf{E}((\overline{\mathbf{T}}_{,}\overline{\mathbf{T}}_{2}^{1}),\\{n-1\\},L),$

so that
$\textup{Shape}(\overline{\mathbf{T}}_{1}^{2})=(\textup{Shape}(\overline{\mathbf{T}})\setminus\\{S_{ij}\\})\cup\\{S_{i-1,j+1}\\}.$
Note that by Proposition 4.2.4,
$n_{v}(\overline{\mathbf{T}}_{1}^{2},\overline{\mathbf{T}}_{2}^{2})\equiv
n_{v}(\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1})\pmod{4}$.
Since
$\mathbf{S}(\overline{\mathbf{T}}_{2}^{2})=\overline{\mathbf{T}}_{2}^{1}$, we
can use Theorem 4.6.2 inductively to find a sequence $\Sigma_{2}$ for
$\Pi^{*}\setminus\\{\alpha_{n}\\}$ such that
$(\overline{\mathbf{T}}_{1}^{2},\overline{\mathbf{T}}_{2}^{2})\in
T_{\Sigma_{2}}^{L}((\overline{\mathbf{T}}_{1}^{1},\overline{\mathbf{T}}_{2}^{1}))$.
Let $(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})\in
T_{\Sigma_{2}}^{L}((\mathbf{T}_{1}^{1},\mathbf{T}_{2}^{1}))$ be such that
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})-L=(\overline{\mathbf{T}}_{1}^{2},\overline{\mathbf{T}}_{2}^{2}).$
Then
$P(n,\mathbf{T}_{1}^{\prime})=P^{\prime}(n,\mathbf{T}_{1}^{1})=P^{\prime}$. By
setting $\Sigma=\Sigma_{1}\Sigma_{2}$, we complete the proof of the lemma in
this case. When $P^{\prime}$ is vertical, a transposed argument applies. This
exhausts all of the possible cases, bringing the proof of the lemma to a
merciful end. ∎

##

The purpose of this section is to describe the generalized $\tau$-invariant in
terms of domino tableaux. For completeness, we begin by recalling its
definition from [5], adapting it to type $D$.

###### 4.7.1. DEFINITION.

Let $X=W$, $\mathcal{S}(n,n)$, or $\mathcal{T}_{D}(n,n)$. We inductively
define a sequence of equivalence relations $\sim_{m}$ on $X$. Consider $x,y\in
X$. We let $x\sim_{0}y$ whenever $\tau^{L}(x)=\tau^{L}(y)$, For $m\geq 1$, we
write $x\sim_{m}y$ if $x\sim_{m-1}y$ and if for every sequence $\Sigma$ for
$\Pi^{*}$ of length one, every element of $T_{\Sigma}^{L}(x)$ is equivalent
under $\sim_{m-1}$ to some element of $T_{\Sigma}^{L}(y)$ and vice-versa. The
generalized $\tau$-invariant $\sim_{GTD}$ is the common refinement of
$\sim_{m}$ for all non-negative integers $m$. Similarly we define
$\sim_{GTDR}$ to be the common refinement of $\sim^{\prime}_{m}$, defined
analogously to $\sim_{m}$, but replacing $\tau^{L}$ by $\tau^{R}$.

As in [5], we extend this definition to $\mathcal{T}_{D}(M_{1},M_{2})$ where
$M_{1}=\\{1,\ldots,n\\}$ in the obvious way. Our main goal for this section is
to establish the following characterization of the generalized
$\tau$-invariant.

###### 4.7.2. THEOREM.

Let
$(\mathbf{T}_{1},\mathbf{T}_{2}),(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})\in\mathcal{T}_{D}(n,n)$.
Then

$(\mathbf{T}_{1},\mathbf{T}_{2})\sim_{GTD}(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})\text{
if and only if
}\mathbf{S}(\mathbf{T}_{1})=\mathbf{S}(\mathbf{T}_{1}^{\prime}).$

We begin by verifying the backwards implication. The following sequence of