$\displaystyle{\mathcal{F}}_{3}$ $\displaystyle=\\{x^{3}+x+1,x^{3}+x^{2}+1\\}$

Thus we expect

  1. $\bullet$

one cuspidal character of $GL_{1}({\mathbb{F}}_{2})$, namely
$\rho_{1}(=\underline{1}_{GL_{1}({\mathbb{F}}_{2})})$,

  2. $\bullet$

one cuspidal character $\rho_{2}$ of $GL_{2}({\mathbb{F}}_{2})$, and

  3. $\bullet$

two cuspidal characters $\rho_{3},\rho_{3}^{\prime}$ of
$GL_{3}({\mathbb{F}}_{2})$.

We will say more about $\rho_{2},\rho_{3},\rho_{3}^{\prime}$ in the next
section.

######  Exercise 4.48.

  * (a)

Show that for $n\geq 2$,

(4.35) $|{\mathcal{C}}_{n}|(=|{\mathcal{F}}_{n}|)=\frac{1}{n}\sum_{d\mid
n}\mu\left(\frac{n}{d}\right)q^{d}$

where $\mu(m)$ is the number-theoretic Möbius function of $m$, that is
$\mu(m)=(-1)^{d}$ if $m=p_{1}\cdots p_{d}$ for $d$ distinct primes, and
$\mu(m)=0$ if $m$ is not squarefree. (Here, the summation sign $\sum_{d\mid
n}$ means a sum over all positive divisors $d$ of $n$.)

  * (b)

Show that (4.35) also counts the necklaces with $n$ beads of $q$ colors
(=equivalence classes under the ${\mathbb{Z}}/n{\mathbb{Z}}$-action of cyclic
rotation on sequences $(a_{1},\ldots,a_{n})$ in ${\mathbb{F}}_{q}^{n}$) which
are primitive in the sense that no nontrivial rotation fixes any of the
sequences within the equivalence class. For example, when $q=2$, here are
representatives of these primitive necklaces for $n=2,3,4$:

$\displaystyle n=2:$ $\displaystyle\\{(0,1)\\}$ $\displaystyle n=3:$
$\displaystyle\\{(0,0,1),(0,1,1)\\}$ $\displaystyle n=4:$
$\displaystyle\\{(0,0,0,1),(0,0,1,1),(0,1,1,1)\\}.$

###  4.7. Steinberg’s unipotent characters

Not surprisingly, the (cuspidal) character $\iota:=\underline{1}_{GL_{1}}$ of
$GL_{1}({\mathbb{F}}_{q})$ plays a distinguished role. The parabolic subgroup
$P_{(1^{n})}$ of $GL_{n}({\mathbb{F}}_{q})$ is the Borel subgroup $B$ of upper
triangular matrices, and we have
$\iota^{n}=\operatorname{Ind}_{B}^{GL_{n}}\underline{1}_{B}={\mathbb{C}}[GL_{n}/B]$
(identifying representations with their characters as usual)186186186Proof.
Exercise 4.31(d) (applied to $G_{*}=GL_{*}$, $\ell=n$,
$\alpha=\left(1^{n}\right)=\left(\underbrace{1,1,\ldots,1}_{n\text{
times}}\right)$ and $\chi_{i}=\iota$) gives
$\iota^{n}=\operatorname{ind}^{n}_{\left(1^{n}\right)}\iota^{\otimes
n}=\underbrace{\operatorname{Ind}^{G_{n}}_{P_{\left(1^{n}\right)}}}_{=\operatorname{Ind}^{GL_{n}}_{B}}\underbrace{\operatorname{Infl}^{P_{\left(1^{n}\right)}}_{G_{\left(1^{n}\right)}}\iota^{\otimes
n}}_{=\underline{1}_{P_{\left(1^{n}\right)}}=\underline{1}_{B}}=\operatorname{Ind}^{GL_{n}}_{B}\underline{1}_{B}={\mathbb{C}}\left[GL_{n}/B\right],$
where the last equality follows from the general fact that if $G$ is a finite
group and $H$ is a subgroup of $G$, then
$\operatorname{Ind}^{G}_{H}\underline{1}_{H}\cong{\mathbb{C}}\left[G/H\right]$
as ${\mathbb{C}}G$-modules.. The subalgebra $A(GL)(\iota)$ of $A(GL)$ is the
${\mathbb{Z}}$-span of the irreducible characters $\sigma$ that appear as
constituents of
$\iota^{n}=\operatorname{Ind}_{B}^{GL_{n}}\underline{1}_{B}={\mathbb{C}}[GL_{n}/B]$
for some $n$.

######  Definition 4.49.

An irreducible character $\sigma$ of $GL_{n}$ appearing as a constituent of
$\operatorname{Ind}_{B}^{GL_{n}}\underline{1}_{B}={\mathbb{C}}[GL_{n}/B]$ is
called a unipotent character. Equivalently, by Frobenius reciprocity, $\sigma$
is unipotent if it contains a nonzero $B$-invariant vector.

In particular, $\underline{1}_{GL_{n}}$ is a unipotent character of $GL_{n}$
for each $n$.

######  Proposition 4.50.

One can choose $\Lambda\cong A(GL)(\iota)$ in Theorem 3.20(g) so that
$h_{n}\longmapsto\underline{1}_{GL_{n}}$.

###### Proof.

Theorem 3.18(a) tells us
$\iota^{2}=\operatorname{Ind}_{B}^{GL_{2}}\underline{1}_{B}$ must have exactly
two irreducible constituents, one of which is $\underline{1}_{GL_{2}}$; call
the other one $\operatorname{St}_{2}$. Choose the isomorphism so as to send
$h_{2}\longmapsto\underline{1}_{GL_{2}}$. Then
$h_{n}\mapsto\underline{1}_{GL_{n}}$ follows from the claim that
$\operatorname{St}_{2}^{\perp}(\underline{1}_{GL_{n}})=0$ for $n\geq 2$: one
has

$\Delta(\underline{1}_{GL_{n}})=\sum_{i+j=n}\left(\operatorname{Res}^{G_{n}}_{P_{i,j}}\underline{1}_{GL_{n}}\right)^{K_{i,j}}=\sum_{i+j=n}\underline{1}_{GL_{i}}\otimes\underline{1}_{GL_{j}}$

so that
$\operatorname{St}_{2}^{\perp}(\underline{1}_{GL_{n}})=(\operatorname{St}_{2},\underline{1}_{GL_{2}})\underline{1}_{GL_{n-2}}=0$
since $\operatorname{St}_{2}\neq\underline{1}_{GL_{2}}$.

∎

This subalgebra $A(GL)(\iota)$, and the unipotent characters
$\chi^{\lambda}_{q}$ corresponding under this isomorphism to the Schur
functions $s_{\lambda}$, were introduced by Steinberg [166]. He wrote down
$\chi^{\lambda}_{q}$ as a virtual sum of induced characters
$\operatorname{Ind}_{P_{\alpha}}^{GL_{n}}\underline{1}_{P_{\alpha}}(=\underline{1}_{G_{\alpha_{1}}}\cdots\underline{1}_{G_{\alpha_{\ell}}})$,
modelled on the Jacobi-Trudi determinantal expression for
$s_{\lambda}=\det(h_{\lambda_{i}-i+j})$. Note that
$\operatorname{Ind}_{P_{\alpha}}^{GL_{n}}\underline{1}_{P_{\alpha}}$ is the
transitive permutation representation ${\mathbb{C}}[G/P_{\alpha}]$ for
$GL_{n}$ permuting the finite partial flag variety $G/P_{\alpha}$, that is,
the set of $\alpha$-flags of subspaces

$\\{0\\}\subset V_{\alpha_{1}}\subset
V_{\alpha_{1}+\alpha_{2}}\subset\cdots\subset
V_{\alpha_{1}+\alpha_{2}+\cdots+\alpha_{\ell-1}}\subset{\mathbb{F}}_{q}^{n}$

where $\dim_{{\mathbb{F}}_{q}}V_{d}=d$ in each case. This character has
dimension equal to $|G/P_{\alpha}|$, with formula given by the $q$-multinomial
coefficient (see e.g. Stanley [164, §1.7]):

$\left[\begin{matrix}n\\\
\alpha\end{matrix}\right]_{q}=\frac{[n]!_{q}}{[\alpha_{1}]!_{q}\cdots[\alpha_{\ell}]!_{q}}$

where $[n]!_{q}:=[n]_{q}[n-1]_{q}\cdots[2]_{q}[1]_{q}$ and
$[n]_{q}:=1+q+\cdots+q^{n-1}=\frac{q^{n}-1}{q-1}$.

Our terminology $\operatorname{St}_{2}$ is motivated by the $n=2$ special case
of the Steinberg character $\operatorname{St}_{n}$, which is the unipotent
character corresponding under the isomorphism in Proposition 4.50 to
$e_{n}=s_{(1^{n})}$. It can be defined by the virtual sum

$\operatorname{St}_{n}:=\chi^{(1^{n})}_{q}=\sum_{\alpha}(-1)^{n-\ell(\alpha)}\operatorname{Ind}_{P_{\alpha}}^{GL_{n}}\underline{1}_{P_{\alpha}}$

in which the sum runs through all compositions $\alpha$ of $n$. This turns out
to be the genuine character for $GL_{n}({\mathbb{F}}_{q})$ acting on the top
homology group of its Tits building: the simplicial complex whose vertices are
nonzero proper subspaces $V$ of ${\mathbb{F}}_{q}^{n}$, and whose simplices
correspond to flags of nested subspaces. One needs to know that this Tits
building has only top homology, so that one can deduce the above character
formula from the Hopf trace formula; see Björner [21].

###  4.8. Examples: $GL_{2}({\mathbb{F}}_{2})$ and $GL_{3}({\mathbb{F}}_{2})$

Let’s get our hands dirty.

######  Example 4.51.

For $n=2$, there are two unipotent characters,
$\chi^{(2)}_{q}=\underline{1}_{GL_{2}}$ and

(4.36)
$\operatorname{St}_{2}:=\chi^{(1,1)}_{q}=\underline{1}_{GL_{1}}^{2}-\underline{1}_{GL_{2}}=\operatorname{Ind}_{B}^{GL_{2}}\underline{1}_{B}-\underline{1}_{GL_{2}}$

since the Jacobi-Trudi formula (2.32) gives
$s_{(1,1)}=\det\left[\begin{matrix}h_{1}&h_{2}\\\
1&h_{1}\end{matrix}\right]=h_{1}^{2}-h_{2}.$ The description (4.36) for this
Steinberg character $\operatorname{St}_{2}$ shows that it has dimension

$|GL_{2}/B|-1=(q+1)-1=q$

and that one can think of it as follows: consider the permutation action of
$GL_{2}$ on the $q+1$ lines $\\{\ell_{0},\ell_{1},\ldots,\ell_{q}\\}$ in the
projective space
${\mathbb{P}}^{1}_{{\mathbb{F}}_{q}}=GL_{2}({\mathbb{F}}_{q})/B$, and take the
invariant subspace perpendicular to the sum of basis elements
$e_{\ell_{0}}+\cdots+e_{\ell_{q}}$.

######  Example 4.52.

Continuing the previous example, but taking $q=2$, we find that we have
constructed two unipotent characters:
$\underline{1}_{GL_{2}}=\chi^{(2)}_{q=2}$ of dimension $1$, and
$\operatorname{St}_{2}=\chi^{(1,1)}_{q=2}$ of dimension $q=2$. This lets us
identify the unique cuspidal character $\rho_{2}$ of
$GL_{2}({\mathbb{F}}_{2})$, using knowledge of the character table of
$GL_{2}({\mathbb{F}}_{2})\cong\mathfrak{S}_{3}$:

|  | $\left[\begin{matrix}1&0\\\ 0&1\end{matrix}\right]$ |  $\left[\begin{matrix}1&1\\\ 0&1\end{matrix}\right]$, $\left[\begin{matrix}1&0\\\ 1&1\end{matrix}\right]$, $\left[\begin{matrix}0&1\\\ 1&0\end{matrix}\right]$ |  $\left[\begin{matrix}1&1\\\ 1&0\end{matrix}\right]$, $\left[\begin{matrix}0&1\\\ 1&1\end{matrix}\right]$
---|---|---|---|---
$\underline{1}_{GL_{2}}=\chi^{(2)}_{q=2}$ | unipotent | 1 | 1 | 1
$\operatorname{St}_{2}=\chi^{(1,1)}_{q=2}$ | unipotent | 2 | 0 | -1
$\rho_{2}$ | cuspidal | 1 | -1 | 1

In other words, the cuspidal character $\rho_{2}$ of
$GL_{2}({\mathbb{F}}_{2})$ corresponds under the isomorphism
$GL_{2}({\mathbb{F}}_{2})\cong\mathfrak{S}_{3}$ to the sign character
$\operatorname{sgn}_{\mathfrak{S}_{3}}$.

######  Example 4.53.

Continuing the previous example to $q=2$ and $n=3$ lets us analyze the
irreducible characters of $GL_{3}({\mathbb{F}}_{2})$. Recalling our labelling
$\rho_{1},\rho_{2},\rho_{3},\rho_{3}^{\prime}$ from Example 4.47 of the
cuspidal characters of $GL_{n}({\mathbb{F}}_{2})$ for $n=1,2,3$, Zelevinsky’s
Theorem 3.12 tells us that the $GL_{3}({\mathbb{F}}_{2})$-irreducible
characters should be labelled by functions
$\\{\rho_{1},\rho_{2},\rho_{3},\rho_{3}^{\prime}\\}\overset{\underline{\lambda}}{\longrightarrow}\operatorname{Par}$
for which

$1\cdot|\underline{\lambda}(\rho_{1})|+2\cdot|\underline{\lambda}(\rho_{2})|+3\cdot|\underline{\lambda}(\rho_{3})|+3\cdot|\underline{\lambda}(\rho^{\prime}_{3})|=3$

We will label such an irreducible character
$\chi^{\underline{\lambda}}=\chi^{(\underline{\lambda}(\rho_{1}),\underline{\lambda}(\rho_{2}),\underline{\lambda}(\rho_{3}),\underline{\lambda}(\rho^{\prime}_{3}))}$.

Three of these irreducibles will be the unipotent characters, mapping under
the isomorphism from Proposition 4.50 as follows:

  1. $\bullet$

$s_{(3)}=h_{3}\longmapsto\chi^{((3),\varnothing,\varnothing,\varnothing)}=\underline{1}_{GL_{3}}$
of dimension $1$.

  2. $\bullet$

$s_{(2,1)}=\det\left[\begin{matrix}h_{2}&h_{3}\\\
1&h_{1}\end{matrix}\right]=h_{2}h_{1}-h_{3}\longmapsto\chi^{((2,1),\varnothing,\varnothing,\varnothing)}=\operatorname{Ind}_{P_{2,1}}^{GL_{3}}\underline{1}_{P_{2,1}}-\underline{1}_{GL_{3}},$

of dimension $\left[\begin{matrix}3\\\
2,1\end{matrix}\right]_{q}-\left[\begin{matrix}3\\\
3\end{matrix}\right]_{q}=[3]_{q}-1=q^{2}+q\overset{q=2}{\rightsquigarrow}6$.

  3. $\bullet$

Lastly,

$\displaystyle s_{(1,1,1)}$
$\displaystyle=\det\left[\begin{matrix}h_{1}&h_{2}&h_{3}\\\ 1&h_{1}&h_{2}\\\
0&1&h_{1}\end{matrix}\right]=h_{1}^{3}-h_{2}h_{1}-h_{1}h_{2}+h_{3}$
$\displaystyle\longmapsto\operatorname{St}_{3}=\chi^{((1,1,1),\varnothing,\varnothing,\varnothing)}=\operatorname{Ind}_{B}^{GL_{3}}\underline{1}_{B}-\operatorname{Ind}_{P_{2,1}}^{GL_{3}}\underline{1}_{P_{2,1}}-\operatorname{Ind}_{P_{1,2}}^{GL_{3}}\underline{1}_{P_{1,2}}+\underline{1}_{GL_{3}}$

of dimension

$\displaystyle\left[\begin{matrix}3\\\
1,1,1\end{matrix}\right]_{q}-\left[\begin{matrix}3\\\
2,1\end{matrix}\right]_{q}-\left[\begin{matrix}3\\\
1,2\end{matrix}\right]_{q}+\left[\begin{matrix}3\\\ 3\end{matrix}\right]_{q}$
$\displaystyle=[3]!_{q}-[3]_{q}-[3]_{q}+1=q^{3}\overset{q=2}{\rightsquigarrow}8.$

There should also be one non-unipotent, non-cuspidal character, namely

$\chi^{((1),(1),\varnothing,\varnothing)}=\rho_{1}\rho_{2}=\operatorname{Ind}_{P_{1,2}}^{GL_{3}}\operatorname{Infl}_{GL_{1}\times
GL_{2}}^{P_{1,2}}\left(\underline{1}_{GL_{1}}\otimes\rho_{2}\right)$

having dimension $\left[\begin{matrix}3\\\ 1,2\end{matrix}\right]_{q}\cdot
1\cdot 1=[3]_{q}\overset{q=2}{\rightsquigarrow}7$.

Finally, we expect cuspidal characters
$\rho_{3}=\chi^{(\varnothing,\varnothing,(1),\varnothing)},\rho_{3}^{\prime}=\chi^{(\varnothing,\varnothing,\varnothing,(1))}$,
whose dimensions $d_{3},d_{3}^{\prime}$ can be deduced from the equation

$1^{2}+6^{2}+8^{2}+7^{2}+d_{3}^{2}+(d_{3}^{\prime})^{2}=|GL_{3}({\mathbb{F}}_{2})|=\left[(q^{3}-q^{0})(q^{3}-q^{1})(q^{3}-q^{2})\right]_{q=2}=168.$

This forces $d_{3}^{2}+(d_{3}^{\prime})^{2}=18$, whose only solution in
positive integers is $d_{3}=d_{3}^{\prime}=3$.

We can check our predictions of the dimensions for the various
$GL_{3}({\mathbb{F}}_{2})$-irreducible characters since
$GL_{3}({\mathbb{F}}_{2})$ is the finite simple group of order $168$ (also
isomorphic to $PSL_{2}({\mathbb{F}}_{7})$), with known character table (see
James and Liebeck [83, p. 318]):
centralizer order 168 8 4 3 7 7 unipotent?/cuspidal?
$\underline{1}_{GL_{3}}=\chi^{((3),\varnothing,\varnothing,\varnothing)}$
unipotent 1 1 1 1 1 1
$\phantom{\operatorname{St}_{3}=}\chi^{((2,1),\varnothing,\varnothing,\varnothing)}$
unipotent 6 2 0 0 -1 -1
$\operatorname{St}_{3}=\chi^{((1,1,1),\varnothing,\varnothing,\varnothing)}$
unipotent 8 0 0 -1 1 1
$\phantom{\operatorname{St}_{3}=}\chi^{((1),(1),\varnothing,\varnothing)}$ 7
-1 -1 1 0 0 $\rho_{3}=\chi^{(\varnothing,\varnothing,(1),\varnothing)}$
cuspidal 3 -1 1 0 $\alpha$ $\overline{\alpha}$
$\rho_{3}^{\prime}=\chi^{(\varnothing,\varnothing,\varnothing,(1))}$ cuspidal
3 -1 1 0 $\overline{\alpha}$ $\alpha$
Here $\alpha:=-1/2+i\sqrt{7}/2$.

######  Remark 4.54.

It is known (see e.g. Bump [28, Cor. 7.4]) that, for $n\geq 2$, the dimension
of any cuspidal irreducible character $\rho$ of $GL_{n}({\mathbb{F}}_{q})$ is

$(q^{n-1}-1)(q^{n-2}-1)\cdots(q^{2}-1)(q-1).$

Note that when $q=2$,

  1. $\bullet$

for $n=2$ this gives $2^{1}-1=1$ for the dimension of $\rho_{2}$, and

  2. $\bullet$

for $n=3$ it gives $(2^{2}-1)(2-1)=3$ for the dimensions of
$\rho_{3},\rho_{3}^{\prime}$,

agreeing with our calculations above. Much more is known about the character
table of $GL_{n}({\mathbb{F}}_{q})$; see Remark 4.68 below, Zelevinsky [183,
Chap. 11], and Macdonald [114, Chap. IV].

###  4.9. The Hall algebra

There is another interesting Hopf subalgebra (and quotient Hopf algebra) of
$A(GL)$, related to unipotent conjugacy classes in $GL_{n}({\mathbb{F}}_{q})$.

######  Definition 4.55.

Say that an element $g$ in $GL_{n}({\mathbb{F}}_{q})$ is unipotent if its
eigenvalues are all equal to $1$. Equivalently, $g\in
GL_{n}({\mathbb{F}}_{q})$ is unipotent if and only if
$g-\operatorname{id}_{{\mathbb{F}}_{q}^{n}}$ is nilpotent. A conjugacy class
in $GL_{n}({\mathbb{F}}_{q})$ is unipotent if its elements are unipotent.

Denote by ${\mathcal{H}}_{n}$ the ${\mathbb{C}}$-subspace of
$R_{\mathbb{C}}(GL_{n})$ consisting of those class functions which are
supported only on unipotent conjugacy classes, and let
${\mathcal{H}}=\bigoplus_{n\geq 0}{\mathcal{H}}_{n}$ as a
${\mathbb{C}}$-subspace of $A_{\mathbb{C}}(GL)=\bigoplus_{n\geq
0}R_{\mathbb{C}}(GL_{n})$.

######  Proposition 4.56.

The subspace ${\mathcal{H}}$ is a Hopf subalgebra of $A_{\mathbb{C}}(GL)$,
which is graded, connected, and of finite type, and self-dual with respect to
the inner product on class functions inherited from $A_{\mathbb{C}}(GL)$. It
is also a quotient Hopf algebra of $A_{\mathbb{C}}(GL)$, as the
${\mathbb{C}}$-linear surjection
$A_{\mathbb{C}}(GL)\twoheadrightarrow{\mathcal{H}}$ restricting class
functions to unipotent classes is a Hopf algebra homomorphism. This surjection
has kernel ${\mathcal{H}}^{\perp}$, which is both an ideal and a two-sided
coideal.

###### Proof.

It is immediately clear that ${\mathcal{H}}^{\perp}$ is a graded
${\mathbb{C}}$-vector subspace of $A_{\mathbb{C}}\left(GL\right)$, whose
$n$-th homogeneous component consists of those class functions on $GL_{n}$
whose values on all unipotent classes are $0$. (This holds no matter whether
the perpendicular space is taken with respect to the Hermitian form
$\left(\cdot,\cdot\right)_{G}$ or with respect to the bilinear form
$\left<\cdot,\cdot\right>_{G}$.) In other words, ${\mathcal{H}}^{\perp}$ is
the kernel of the surjection
$A_{\mathbb{C}}(GL)\twoheadrightarrow{\mathcal{H}}$ defined in the
proposition.

Given two class functions $\chi_{i},\chi_{j}$ on $GL_{i},GL_{j}$ and $g$ in
$GL_{i+j}$, one has

(4.37)
$\left(\chi_{i}\cdot\chi_{j}\right)(g)=\frac{1}{|P_{i,j}|}\sum_{\begin{subarray}{c}h\in
GL_{i+j}:\\\ h^{-1}gh=\left[\begin{matrix}g_{i}&*\\\
0&g_{j}\end{matrix}\right]\in
P_{i,j}\end{subarray}}\chi_{i}(g_{i})\chi_{j}(g_{j}).$

Since $g$ is unipotent if and only if $h^{-1}gh$ is unipotent if and only if
both $g_{i},g_{j}$ are unipotent, the formula (4.37) shows both that
${\mathcal{H}}$ is a subalgebra187187187Indeed, if $\chi_{i}$ and $\chi_{j}$
are both supported only on unipotent classes, then the same holds for
$\chi_{i}\cdot\chi_{j}$.  and that ${\mathcal{H}}^{\perp}$ is a two-sided
ideal188188188In fact, if one of $\chi_{i}$ and $\chi_{j}$ annihilates all
unipotent classes, then so does $\chi_{i}\cdot\chi_{j}$.. It also shows that
the surjection $A_{\mathbb{C}}(GL)\twoheadrightarrow{\mathcal{H}}$ restricting
every class function to unipotent classes is an algebra
homomorphism189189189because if $g$ is unipotent, then the only values of
$\chi_{i}$ and $\chi_{j}$ appearing on the right hand side of (4.37) are those
on unipotent elements.

Similarly, for class functions $\chi$ on $GL_{n}$ and $(g_{i},g_{j})$ in
$GL_{i,j}=GL_{i}\times GL_{j}$, one has

$\Delta(\chi)(g_{i},g_{j})=\frac{1}{q^{ij}}\sum_{k\in{\mathbb{F}}_{q}^{i\times
j}}\chi\left[\begin{matrix}g_{i}&k\\\ 0&g_{j}\end{matrix}\right]$

using (4.13). This shows both that ${\mathcal{H}}$ is a sub-coalgebra of
$A=A_{\mathbb{C}}(GL)$ (that is, it satisfies
$\Delta{\mathcal{H}}\subset{\mathcal{H}}\otimes{\mathcal{H}}$) and that
${\mathcal{H}}^{\perp}$ is a two-sided coideal (that is, we have
$\Delta({\mathcal{H}}^{\perp})\subset{\mathcal{H}}^{\perp}\otimes
A+A\otimes{\mathcal{H}}^{\perp}$), since it shows that if $\chi$ is supported
only on unipotent classes, then $\Delta(\chi)$ vanishes on $(g_{1},g_{2})$
that have either $g_{1}$ or $g_{2}$ non-unipotent. It also shows that the
surjection $A_{\mathbb{C}}(GL)\twoheadrightarrow{\mathcal{H}}$ restricting
every class function to unipotent classes is a coalgebra homomorphism. The
rest follows. ∎

The subspace ${\mathcal{H}}$ is called the Hall algebra. It has an obvious
orthogonal ${\mathbb{C}}$-basis, with interesting structure constants.

######  Definition 4.57.

Given a partition $\lambda$ of $n$, let $J_{\lambda}$ denote the
$GL_{n}$-conjugacy class of unipotent matrices whose Jordan type (that is, the
list of the sizes of the Jordan blocks, in decreasing order) is given by
$\lambda$. Furthermore, let $z_{\lambda}(q)$ denote the size of the
centralizer of any element of this conjugacy class $J_{\lambda}$.

The indicator class functions190190190Here we use the following notation:
Whenever $P$ is a subset of a group $G$, we denote by $\underline{1}_{P}$ the
map $G\to{\mathbb{C}}$ which sends every element of $P$ to $1$ and all
remaining elements of $G$ to $0$. This is not in conflict with the notation
$\underline{1}_{G}$ for the trivial character of $G$, since
$\underline{1}_{P}=\underline{1}_{G}$ for $P=G$. Note that $\underline{1}_{P}$
is a class function when $P$ is a union of conjugacy classes of $G$.
$\\{\underline{1}_{J_{\lambda}}\\}_{\lambda\in\operatorname{Par}}$ form a
${\mathbb{C}}$-basis for ${\mathcal{H}}$ whose multiplicative structure
constants are called the Hall coefficients $g^{\lambda}_{\mu,\nu}(q)$:

$\underline{1}_{J_{\mu}}\underline{1}_{J_{\nu}}=\sum_{\lambda}g^{\lambda}_{\mu,\nu}(q)\,\,\underline{1}_{J_{\lambda}}.$

Because the dual basis to $\\{\underline{1}_{J_{\lambda}}\\}$ is
$\\{z_{\lambda}(q)\underline{1}_{J_{\lambda}}\\}$, self-duality of
${\mathcal{H}}$ shows that the Hall coefficients are (essentially) also
structure constants for the comultiplication:

$\Delta\underline{1}_{J_{\lambda}}=\sum_{\mu,\nu}g^{\lambda}_{\mu,\nu}(q)\frac{z_{\mu}(q)z_{\nu}(q)}{z_{\lambda}(q)}\cdot\underline{1}_{J_{\mu}}\otimes\underline{1}_{J_{\nu}}.$

The Hall coefficient $g^{\lambda}_{\mu,\nu}(q)$ has the following
interpretation.

######  Proposition 4.58.

Fix any $g$ in $GL_{n}({\mathbb{F}}_{q})$ acting unipotently on
${\mathbb{F}}_{q}^{n}$ with Jordan type $\lambda$. Then
$g^{\lambda}_{\mu,\nu}(q)$ counts the $g$-stable ${\mathbb{F}}_{q}$-subspaces
$V\subset{\mathbb{F}}_{q}^{n}$ for which the restriction $g|V$ acts with
Jordan type $\mu$, and the induced map $\bar{g}$ on the quotient space
${\mathbb{F}}_{q}^{n}/V$ has Jordan type $\nu$.

###### Proof.

Given $\mu,\nu$ partitions of $i,j$ with $i+j=n$, taking $\chi_{i},\chi_{j}$
equal to $\underline{1}_{J_{\mu}},\underline{1}_{J_{\nu}}$ in (4.37) shows
that for any $g$ in $GL_{n}$, the value of
$\left(\underline{1}_{J_{\mu}}\cdot\underline{1}_{J_{\nu}}\right)(g)$ is given
by

(4.38) $\frac{1}{|P_{i,j}|}\left|\left\\{h\in
GL_{n}:h^{-1}gh=\left[\begin{matrix}g_{i}&*\\\
0&g_{j}\end{matrix}\right]\text{ with }g_{i}\in J_{\mu},g_{j}\in
J_{\nu}\right\\}\right|.$

Let $S$ denote the set appearing in (4.38), and let ${\mathbb{F}}_{q}^{i}$
denote the $i$-dimensional subspace of ${\mathbb{F}}_{q}^{n}$ spanned by the
first $i$ standard basis vectors. Note that the condition on an element $h$ in
$S$ saying that $h^{-1}gh$ is in block upper-triangular form can be re-
expressed by saying that the subspace $V:=h({\mathbb{F}}_{q}^{i})$ is
$g$-stable. One then sees that the map
$h\overset{\varphi}{\longmapsto}V=h({\mathbb{F}}_{q}^{i})$ surjects $S$ onto
the set of $i$-dimensional $g$-stable subspaces $V$ of ${\mathbb{F}}_{q}^{n}$
for which $g|V$ and $\bar{g}$ are unipotent of types $\mu,\nu$, respectively.
Furthermore, for any particular such $V$, its fiber $\varphi^{-1}(V)$ in $S$
is a coset of the stabilizer within $GL_{n}$ of $V$, which is conjugate to
$P_{i,j}$, and hence has cardinality $|\varphi^{-1}(V)|=|P_{i,j}|$. This
proves the assertion of the proposition. ∎

The Hall algebra ${\mathcal{H}}$ will turn out to be isomorphic to the ring
$\Lambda_{\mathbb{C}}$ of symmetric functions with ${\mathbb{C}}$
coefficients, via a composite $\varphi$ of three maps

$\Lambda_{\mathbb{C}}\longrightarrow A(GL)(\iota)_{\mathbb{C}}\longrightarrow
A(GL)_{\mathbb{C}}\longrightarrow{\mathcal{H}}$

in which the first map is the isomorphism from Proposition 4.50, the second is
inclusion, and the third is the quotient map from Proposition 4.56.

######  Theorem 4.59.

The above composite $\varphi$ is a Hopf algebra isomorphism, sending

$\begin{array}[]{rcl}h_{n}&\longmapsto&\sum_{\lambda\in\operatorname{Par}_{n}}\underline{1}_{J_{\lambda}},\\\
e_{n}&\longmapsto&q^{\binom{n}{2}}\underline{1}_{J_{(1^{n})}},\\\
p_{n}&\longmapsto&\sum_{\lambda\in\operatorname{Par}_{n}}(q;q)_{\ell(\lambda)-1}\underline{1}_{J_{\lambda}}\qquad\left(\text{for
}n>0\right),\end{array}$

where

$(x;q)_{m}:=(1-x)(1-qx)(1-q^{2}x)\cdots(1-q^{m-1}x)\qquad\text{ for all
}m\in{\mathbb{N}}.$

###### Proof.

That $\varphi$ is a graded Hopf morphism follows because it is a composite of
three such morphisms. We claim that once one shows the formula for the
(nonzero) image $\varphi(p_{n})$ given above is correct, then this will
already show $\varphi$ is an isomorphism, by the following argument. Note
first that $\Lambda_{\mathbb{C}}$ and ${\mathcal{H}}$ both have dimension
$|\operatorname{Par}_{n}|$ for their $n^{th}$ homogeneous components, so it
suffices to show that the graded map $\varphi$ is injective. On the other
hand, both $\Lambda_{\mathbb{C}}$ and ${\mathcal{H}}$ are (graded, connected,
finite type) self-dual Hopf algebras (although with respect to a sesquilinear
form), so Theorem 3.8 says that each is the symmetric algebra on its space of
primitive elements. Thus it suffices to check that $\varphi$ is injective when
restricted to their subspaces of primitives.191191191An alternative way to see
that it suffices to check this is by recalling Exercise 1.54(c). For
$\Lambda_{\mathbb{C}}$, by Corollary 3.9 the primitives are spanned by
$\\{p_{1},p_{2},\ldots\\}$, with only one basis element in each degree $n\geq
1$. Hence $\varphi$ is injective on the subspace of primitives if and only if
it does not annihilate any $p_{n}$.

Thus it only remains to show the above formulas for the images of
$h_{n},e_{n},p_{n}$ under $\varphi$. This is clear for $h_{n}$, since
Proposition 4.50 shows that it maps under the first two composites to the
indicator function $\underline{1}_{GL_{n}}$ which then restricts to the sum of
indicators
$\sum_{\lambda\in\operatorname{Par}_{n}}\underline{1}_{J_{\lambda}}$ in
${\mathcal{H}}$. For $e_{n},p_{n}$, we resort to generating functions. Let
$\tilde{h}_{n},\tilde{e}_{n},\tilde{p}_{n}$ denote the three putative images
in ${\mathcal{H}}$ of $h_{n},e_{n},p_{n}$, appearing on the right side in the
theorem, and define generating functions

$\tilde{H}(t):=\sum_{n\geq 0}\tilde{h}_{n}t^{n},\quad\tilde{E}(t):=\sum_{n\geq
0}\tilde{e}_{n}t^{n},\quad\tilde{P}(t):=\sum_{n\geq
0}\tilde{p}_{n+1}t^{n}\qquad\qquad\text{ in }{\mathcal{H}}[[t]].$

We wish to show that the map
$\varphi[[t]]:\Lambda_{\mathbb{C}}[[t]]\to{\mathcal{H}}[[t]]$ (induced by
$\varphi$) maps $H(t),E(t),P(t)$ in $\Lambda[[t]]$ to these three generating
functions192192192See (2.24), (2.25), (2.40) for the definitions of
$H(t),E(t),P(t)$.. Since we have already shown this is correct for $H(t)$, by
(2.26), (2.40), it suffices to check that in ${\mathcal{H}}[[t]]$ one has

$\begin{array}[]{rccl}\tilde{H}(t)\tilde{E}(-t)=1,&\text{ or equivalently,
}&\sum_{k=0}^{n}(-1)^{k}\tilde{e}_{k}\tilde{h}_{n-k}=\delta_{0,n};\\\
\tilde{H}^{\prime}(t)\tilde{E}(-t)=\tilde{P}(t),&\text{ or equivalently,
}&\sum_{k=0}^{n}(-1)^{k}(n-k)\tilde{e}_{k}\tilde{h}_{n-k}=\tilde{p}_{n}.\end{array}$

Thus it would be helpful to evaluate the class function
$\tilde{e}_{k}\tilde{h}_{n-k}$. Note that a unipotent $g$ in $GL_{n}$ having
$\ell$ Jordan blocks has an $\ell$-dimensional $1$-eigenspace, so that the
number of $k$-dimensional $g$-stable ${\mathbb{F}}_{q}$-subspaces of
${\mathbb{F}}_{q}^{n}$ on which $g$ has Jordan type $\left(1^{k}\right)$ (that
is, on which $g$ acts as the identity) is the $q$-binomial coefficient

$\left[\begin{matrix}\ell\\\
k\end{matrix}\right]_{q}=\frac{(q;q)_{\ell}}{(q;q)_{k}(q;q)_{\ell-k}},$

counting $k$-dimensional ${\mathbb{F}}_{q}$-subspaces $V$ of an
$\ell$-dimensional ${\mathbb{F}}_{q}$-vector space; see, e.g., [164, §1.7].
Hence, for a unipotent $g$ in $GL_{n}$ having $\ell$ Jordan blocks, we have

$(\tilde{e}_{k}\tilde{h}_{n-k})(g)=q^{\binom{k}{2}}\cdot\left(\underline{1}_{J_{(1^{k})}}\cdot\tilde{h}_{n-k}\right)(g)=q^{\binom{k}{2}}\cdot\sum_{\nu\in\operatorname{Par}_{n-k}}\left(\underline{1}_{J_{(1^{k})}}\cdot\underline{1}_{J_{\nu}}\right)(g)=q^{\binom{k}{2}}\left[\begin{matrix}\ell\\\
k\end{matrix}\right]_{q}$

(by Proposition 4.58). Thus one needs for $\ell\geq 1$ that

(4.39)
$\displaystyle\sum_{k=0}^{\ell}(-1)^{k}q^{\binom{k}{2}}\left[\begin{matrix}\ell\\\
k\end{matrix}\right]_{q}$ $\displaystyle=0,$ (4.40)
$\displaystyle\sum_{k=0}^{\ell}(-1)^{k}(n-k)q^{\binom{k}{2}}\left[\begin{matrix}\ell\\\
k\end{matrix}\right]_{q}$ $\displaystyle=(q;q)_{\ell-1}.$

Identity (4.39) comes from setting $x=1$ in the $q$-binomial theorem [164,
Exer. 3.119]:

$\sum_{k=0}^{\ell}(-1)^{k}q^{\binom{k}{2}}\left[\begin{matrix}\ell\\\
k\end{matrix}\right]_{q}x^{\ell-k}=(x-1)(x-q)(x-q^{2})\cdots(x-q^{\ell-1}).$

Identity (4.40) comes from taking $\frac{d}{dx}$ in the $q$-binomial theorem,
then setting $x=1$, and finally adding $(n-\ell)$ times (4.39). ∎

######  Exercise 4.60.

Fix a prime power $q$. For any $k\in{\mathbb{N}}$, and any $k$ partitions
$\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(k)}$, we define a family
$\left(g_{\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(k)}}^{\lambda}\left(q\right)\right)_{\lambda\in\operatorname{Par}}$
of elements of ${\mathbb{C}}$ by the equation

$\underline{1}_{J_{\lambda^{(1)}}}\underline{1}_{J_{\lambda^{(2)}}}\cdots\underline{1}_{J_{\lambda^{(k)}}}=\sum_{\lambda\in\operatorname{Par}}g_{\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(k)}}^{\lambda}\left(q\right)\underline{1}_{J_{\lambda}}$

in ${\mathcal{H}}$. This notation generalizes the notation
$g^{\lambda}_{\mu,\nu}\left(q\right)$ we introduced in Definition 4.57. Note
that $g^{\lambda}_{\mu}\left(q\right)=\delta_{\lambda,\mu}$ for any two
partitions $\lambda$ and $\mu$, and that
$g^{\lambda}\left(q\right)=\delta_{\lambda,\varnothing}$ for any partition
$\lambda$ (where $g^{\lambda}\left(q\right)$ is to be understood as
$g_{\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(k)}}^{\lambda}\left(q\right)$
for $k=0$).

  * (a)

Let $\lambda\in\operatorname{Par}$, and let $n=\left|\lambda\right|$. Let $V$
be an $n$-dimensional ${\mathbb{F}}_{q}$-vector space, and let $g$ be a
unipotent endomorphism of $V$ having Jordan type $\lambda$. Let
$k\in{\mathbb{N}}$, and let $\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(k)}$
be $k$ partitions. A
$\left(\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(k)}\right)$-compatible
$g$-flag will mean a sequence $0=V_{0}\subset V_{1}\subset
V_{2}\subset\cdots\subset V_{k}=V$ of $g$-invariant ${\mathbb{F}}_{q}$-vector
subspaces $V_{i}$ of $V$ such that for every
$i\in\left\\{1,2,\ldots,k\right\\}$, the endomorphism of $V_{i}/V_{i-1}$
induced by $g$ 193193193This is well-defined. In fact, both $V_{i}$ and
$V_{i-1}$ are $g$-invariant, so that $g$ restricts to an endomorphism of
$V_{i}$, which further restricts to an endomorphism of $V_{i-1}$, and thus
gives rise to an endomorphism of $V_{i}/V_{i-1}$. has Jordan type
$\lambda^{(i)}$.

Show that
$g_{\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(k)}}^{\lambda}\left(q\right)$
is the number of
$\left(\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(k)}\right)$-compatible
$g$-flags.194194194This can be seen as a generalization of Proposition 4.58.
In fact, if $\mu$ and $\nu$ are two partitions, then a
$\left(\mu,\nu\right)$-compatible $g$-flag is a sequence $0=V_{0}\subset
V_{1}\subset V_{2}=V$ of $g$-invariant ${\mathbb{F}}_{q}$-vector subspaces
$V_{i}$ of $V$ such that the endomorphism of $V_{1}/V_{0}\cong V_{1}$ induced
by $g$ has Jordan type $\mu$, and the endomorphism of $V_{2}/V_{1}\cong
V/V_{1}$ induced by $g$ has Jordan type $\nu$. Choosing such a sequence
amounts to choosing $V_{1}$ (since there is only one choice for each of
$V_{0}$ and $V_{2}$), and the conditions on this $V_{1}$ are precisely the
conditions on $V$ in Proposition 4.58.

  * (b)

Let $\lambda\in\operatorname{Par}$. Let $k\in{\mathbb{N}}$, and let
$\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(k)}$ be $k$ partitions. Show
that
$g_{\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(k)}}^{\lambda}\left(q\right)=0$
unless
$\left|\lambda^{(1)}\right|+\left|\lambda^{(2)}\right|+\cdots+\left|\lambda^{(k)}\right|=\left|\lambda\right|$
and $\lambda^{(1)}+\lambda^{(2)}+\cdots+\lambda^{(k)}\triangleright\lambda$.
(Here and in the following, we are using the notations of Exercise 2.94).

  * (c)

Let $\lambda\in\operatorname{Par}$, and let us write the transpose partition
$\lambda^{t}$ as
$\lambda^{t}=\left(\left(\lambda^{t}\right)_{1},\left(\lambda^{t}\right)_{2},\ldots,\left(\lambda^{t}\right)_{\ell}\right)$.
Show that
$g_{\left(1^{\left(\lambda^{t}\right)_{1}}\right),\left(1^{\left(\lambda^{t}\right)_{2}}\right),\ldots,\left(1^{\left(\lambda^{t}\right)_{\ell}}\right)}^{\lambda}\left(q\right)\neq
0$.

  * (d)

Let $n\in{\mathbb{N}}$ and $\lambda\in\operatorname{Par}_{n}$. Show that

$\varphi\left(e_{\lambda}\right)=\sum_{\mu\in\operatorname{Par}_{n};\
\lambda^{t}\triangleright\mu}\alpha_{\lambda,\mu}\underline{1}_{J_{\mu}}$

for some coefficients $\alpha_{\lambda,\mu}\in{\mathbb{C}}$ satisfying
$\alpha_{\lambda,\lambda^{t}}\neq 0$.

  * (e)

Give another proof of the fact that the map $\varphi$ is injective.

[Hint: For (b), use Exercise 2.99(b).]

We next indicate, without proof, how ${\mathcal{H}}$ relates to the classical
Hall algebra.

######  Definition 4.61.

Let $p$ be a prime. The usual Hall algebra, or what Schiffmann [150, §2.3]
calls Steinitz’s classical Hall algebra (see also Macdonald [114, Chap. II]),
has ${\mathbb{Z}}$-basis elements
$\\{u_{\lambda}\\}_{\lambda\in\operatorname{Par}}$, with the multiplicative
structure constants $g^{\lambda}_{\mu,\nu}(p)$ in

$u_{\mu}u_{\nu}=\sum_{\lambda}g^{\lambda}_{\mu,\nu}(p)\,\,u_{\lambda}$

defined as follows: fix a finite abelian $p$-group $L$ of type $\lambda$,
meaning that

$L\cong\bigoplus_{i=1}^{\ell(\lambda)}{\mathbb{Z}}/p^{\lambda_{i}}{\mathbb{Z}},$

and let $g^{\lambda}_{\mu,\nu}(p)$ be the number of subgroups $M$ of $L$ of
type $\mu$, for which the quotient $N:=L/M$ is of type $\nu$. In other words,
$g^{\lambda}_{\mu,\nu}(p)$ counts, for a fixed abelian $p$-group $L$ of type
$\lambda$, the number of short exact sequences $0\rightarrow M\rightarrow
L\rightarrow N\rightarrow 0$ in which $M,N$ have types $\mu,\nu$, respectively
(modulo isomorphism of short exact sequences restricting to the identity on
$L$).

We claim that when one takes the finite field ${\mathbb{F}}_{q}$ of order
$q=p$ a prime, the ${\mathbb{Z}}$-linear map

(4.41) $u_{\lambda}\longmapsto\underline{1}_{J_{\lambda}}$

gives an isomorphism from this classical Hall algebra to the
${\mathbb{Z}}$-algebra ${\mathcal{H}}_{\mathbb{Z}}\subset{\mathcal{H}}$. The
key point is Hall’s Theorem, a non-obvious statement for which Macdonald
includes two proofs in [114, Chap. II], one of them due to
Zelevinsky195195195See also [150, Thm. 2.6, Prop. 2.7] for quick proofs of
part of it, similar to Zelevinsky’s. Another proof, based on a recent
category-theoretical paradigm, can be found in [49, Theorem 3.53].. To state
it, we first recall some notions about discrete valuation rings.

######  Definition 4.62.

A discrete valuation ring (DVR) $\mathfrak{o}$ is a principal ideal domain
having only one maximal ideal $\mathfrak{m}\neq 0$, with quotient
$k=\mathfrak{o}/\mathfrak{m}$ called its residue field.

The structure theorem for finitely generated modules over a PID implies that
an $\mathfrak{o}$-module $L$ with finite composition series of composition
length $n$ must have
$L\cong\bigoplus_{i=1}^{\ell(\lambda)}\mathfrak{o}/\mathfrak{m}^{\lambda_{i}}$
for some partition $\lambda$ of $n$; say $L$ has type  $\lambda$ in this
situation.

Here are the two crucial examples for us.

######  Example 4.63.

For any field ${\mathbb{F}}$, the power series ring
$\mathfrak{o}={\mathbb{F}}[[t]]$ is a DVR with maximal ideal
$\mathfrak{m}=(t)$ and residue field
$k=\mathfrak{o}/\mathfrak{m}={\mathbb{F}}[[t]]/(t)\cong{\mathbb{F}}$. An
$\mathfrak{o}$-module $L$ of type $\lambda$ is an ${\mathbb{F}}$-vector space
together with an ${\mathbb{F}}$-linear transformation
$T\in\operatorname{End}L$ that acts on $L$ nilpotently (so that $g:=T+1$ acts
unipotently, where $1=\operatorname{id}_{L}$) with Jordan blocks of sizes
given by $\lambda$: each summand
$\mathfrak{o}/\mathfrak{m}^{\lambda_{i}}={\mathbb{F}}[[t]]/(t^{\lambda_{i}})$
of $L$ has an ${\mathbb{F}}$-basis $\\{1,t,t^{2},\ldots,t^{\lambda_{i}-1}\\}$
on which the map $T$ that multiplies by $t$ acts as a nilpotent Jordan block
of size $\lambda_{i}$. Note also that, in this setting,
$\mathfrak{o}$-submodules are the same as $T$-stable (or $g$-stable)
${\mathbb{F}}$-subspaces.

######  Example 4.64.

The ring of $p$-adic integers $\mathfrak{o}={\mathbb{Z}}_{p}$ is a DVR with
maximal ideal $\mathfrak{m}=(p)$ and residue field
$k=\mathfrak{o}/\mathfrak{m}={\mathbb{Z}}_{p}/p{\mathbb{Z}}_{p}\cong{\mathbb{Z}}/p{\mathbb{Z}}$.
An $\mathfrak{o}$-module $L$ of type $\lambda$ is an abelian $p$-group of type
$\lambda$: for each summand,
$\mathfrak{o}/\mathfrak{m}^{\lambda_{i}}={\mathbb{Z}}_{p}/p^{\lambda_{i}}{\mathbb{Z}}_{p}\cong{\mathbb{Z}}/p^{\lambda_{i}}{\mathbb{Z}}$.
Note also that, in this setting, $\mathfrak{o}$-submodules are the same as
subgroups.

One last notation: $n(\lambda):=\sum_{i\geq 1}(i-1)\lambda_{i}$, for $\lambda$
in $\operatorname{Par}$. Hall’s Theorem is as follows.

######  Theorem 4.65.

Assume $\mathfrak{o}$ is a DVR with maximal ideal $\mathfrak{m}$, and that its
residue field $k=\mathfrak{o}/\mathfrak{m}$ is finite of cardinality $q$. Fix
an $\mathfrak{o}$-module $L$ of type $\lambda$. Then the number of
$\mathfrak{o}$-submodules $M$ of type $\mu$ for which the quotient $N=L/M$ is
of type $\nu$ can be written as the specialization

$[g^{\lambda}_{\mu,\nu}(t)]_{t=q}$

of a polynomial $g^{\lambda}_{\mu,\nu}(t)$ in ${\mathbb{Z}}[t]$, called the
Hall polynomial.

Furthermore, the Hall polynomial $g^{\lambda}_{\mu,\nu}(t)$ has degree at most
$n(\lambda)-(n(\mu)+n(\nu))$, and its coefficient of
$t^{n(\lambda)-(n(\mu)+n(\nu))}$ is the Littlewood-Richardson coefficient
$c^{\lambda}_{\mu,\nu}$.

Comparing what Hall’s Theorem says in Examples 4.63 and 4.64, shows that the
map (4.41) gives the desired isomorphism from the classical Hall algebra to
${\mathcal{H}}_{\mathbb{Z}}$.

We close this section with some remarks on the vast literature on Hall
algebras that we will not discuss here.

######  Remark 4.66.

Macdonald’s version of Hall’s Theorem [114, (4.3)] is stronger than Theorem
4.65, and useful for certain applications: he shows that
$g^{\lambda}_{\mu,\nu}(t)$ is the zero polynomial whenever the Littlewood-
Richardson coefficient $c^{\lambda}_{\mu,\nu}$ is zero.

######  Remark 4.67.

In general, not all coefficients of the Hall polynomials
$g^{\lambda}_{\mu,\nu}(t)$ are nonnegative (see Butler/Hales [30] for a study
of when they are); it often happens that $g^{\lambda}_{\mu,\nu}(1)=0$ despite
$g^{\lambda}_{\mu,\nu}(t)$ not being the zero polynomial196196196Actually,
Butler/Hales show in [30, proof of Prop. 2.4] that the values
$g^{\lambda}_{\mu,\nu}(1)$ are the structure constants of the ring $\Lambda$
with respect to its basis
$\left(m_{\lambda}\right)_{\lambda\in\operatorname{Par}}$: we have
$m_{\mu}m_{\nu}=\sum_{\lambda\in\operatorname{Par}}g^{\lambda}_{\mu,\nu}(1)m_{\lambda}$
for all partitions $\mu$ and $\nu$.. However, in [88, Thm. 4.2], Klein showed
that the polynomial values $g^{\lambda}_{\mu,\nu}\left(p\right)$ for $p$ prime
are always positive when $c^{\lambda}_{\mu,\nu}\neq 0$. (This easily yields
the same result for $p$ a prime power.)

######  Remark 4.68.

Zelevinsky in [183, Chaps 10, 11] uses the isomorphism
$\Lambda_{\mathbb{C}}\rightarrow{\mathcal{H}}$ to derive J. Green’s formula
for the value of any irreducible character $\chi$ of $GL_{n}$ on any unipotent
class $J_{\lambda}$. The answer involves values of irreducible characters of
$\mathfrak{S}_{n}$ along with Green’s polynomials $Q^{\lambda}_{\mu}(q)$ (see
Macdonald [114, §III.7]; they are denoted $Q(\lambda,\mu)$ by Zelevinsky),
which express the images under the isomorphism of Theorem 4.59 of the
symmetric function basis $\\{p_{\mu}\\}$ in terms of the basis
$\\{\underline{1}_{J_{\lambda}}\\}$.

######  Remark 4.69.

The Hall polynomials $g^{\lambda}_{\mu,\nu}(t)$ also essentially give the
multiplicative structure constants for $\Lambda({\mathbf{x}})[t]$ with respect
to its basis of Hall-Littlewood symmetric functions
$P_{\lambda}=P_{\lambda}({\mathbf{x}};t)$:

$P_{\mu}P_{\nu}=\sum_{\lambda}t^{n(\lambda)-(n(\mu)+n(\nu))}g^{\lambda}_{\mu,\nu}(t^{-1})P_{\lambda}.$

See Macdonald [114, §III.3].

######  Remark 4.70.

Schiffmann [150] discusses self-dual Hopf algebras which vastly generalize the
classical Hall algebra called Ringel-Hall algebras, associated to abelian
categories which are hereditary. Examples come from categories of nilpotent
representations of quivers; the quiver having exactly one node and one arc
recovers the classical Hall algebra ${\mathcal{H}}_{\mathbb{Z}}$ discussed
above.

######  Remark 4.71.

The general linear groups $GL_{n}({\mathbb{F}}_{q})$ are one of four families
of so-called classical groups. Progress has been made on extending
Zelevinsky’s PSH theory to the other families:

(a) Work of Thiem and Vinroot [174] shows that the tower $\\{G_{*}\\}$ of
finite unitary groups $U_{n}({\mathbb{F}}_{q^{2}})$ give rise to another
positive self-dual Hopf algebra $A=\bigoplus_{n\geq
0}R(U_{n}({\mathbb{F}}_{q^{2}}))$, in which the role of Harish-Chandra
induction is played by Deligne-Lusztig induction. In this theory, character
and degree formulas for $U_{n}({\mathbb{F}}_{q^{2}})$ are related to those of
$GL_{n}({\mathbb{F}}_{q})$ by substituting $q\mapsto-q$, along with
appropriate scalings by $\pm 1$, a phenomenon sometimes called Ennola duality.
See also [165, §4].

(b) van Leeuwen [102] has studied $\bigoplus_{n\geq
0}R\left(Sp_{2n}\left({\mathbb{F}}_{q}\right)\right)$, $\bigoplus_{n\geq
0}R\left(O_{2n}\left({\mathbb{F}}_{q}\right)\right)$ and $\bigoplus_{n\geq
0}R\left(U_{n}\left({\mathbb{F}}_{q^{2}}\right)\right)$ not as Hopf algebras,
but rather as so-called twisted PSH-modules over the PSH $A(GL)$ (a “deformed”
version of the older notion of Hopf modules). He classified these PSH-modules
axiomatically similarly to Zelevinsky’s above classification of PSH’s.

(c) In a recent honors thesis [159], Shelley-Abrahamson defined yet another
variation of the concept of Hopf modules, named $2$-compatible Hopf modules,
and identified $\bigoplus_{n\geq
0}R\left(Sp_{2n}\left({\mathbb{F}}_{q}\right)\right)$ and $\bigoplus_{n\geq
0}R\left(O_{2n+1}\left({\mathbb{F}}_{q}\right)\right)$ as such modules over
$A(GL)$.

##  5\. Quasisymmetric functions and $P$-partitions

We discuss here our next important example of a Hopf algebra arising in
combinatorics: the quasisymmetric functions of Gessel [63], with roots in work
of Stanley [161] on $P$-partitions.

###  5.1. Definitions, and Hopf structure

The definitions of quasisymmetric functions require a totally ordered variable
set. Usually we will use a variable set denoted
${\mathbf{x}}=(x_{1},x_{2},\ldots)$ with the usual ordering
$x_{1}<x_{2}<\ldots$. However, it is good to have some flexibility in changing
the ordering, which is why we make the following definition.

######  Definition 5.1.

Given any totally ordered set $I$, create a totally ordered variable set
$\\{x_{i}\\}_{i\in I}$, and then let $R(\\{x_{i}\\}_{i\in I})$ denote the
power series of bounded degree in $\\{x_{i}\\}_{i\in I}$ having coefficients
in $\mathbf{k}$.

The quasisymmetric functions $\operatorname{QSym}(\\{x_{i}\\}_{i\in I})$ over
the alphabet $(\\{x_{i}\\}_{i\in I})$ will be the $\mathbf{k}$-submodule
consisting of the elements $f$ in $R(\\{x_{i}\\}_{i\in I})$ that have the same
coefficient on the monomials $x_{i_{1}}^{\alpha_{1}}\cdots
x_{i_{\ell}}^{\alpha_{\ell}}$ and $x_{j_{1}}^{\alpha_{1}}\cdots
x_{j_{\ell}}^{\alpha_{\ell}}$ whenever both $i_{1}<\cdots<i_{\ell}$ and
$j_{1}<\cdots<j_{\ell}$ in the total order on $I$. We write
$\operatorname{QSym}_{\mathbf{k}}(\\{x_{i}\\}_{i\in I})$ instead of
$\operatorname{QSym}(\\{x_{i}\\}_{i\in I})$ to stress the choice of base ring
$\mathbf{k}$.

It immediately follows from this definition that
$\operatorname{QSym}(\\{x_{i}\\}_{i\in I})$ is a free $\mathbf{k}$-submodule
of $R(\\{x_{i}\\}_{i\in I})$, having as $\mathbf{k}$-basis elements the
monomial quasisymmetric functions

$M_{\alpha}(\\{x_{i}\\}_{i\in I}):=\sum_{i_{1}<\cdots<i_{\ell}\text{ in
}I}x_{i_{1}}^{\alpha_{1}}\cdots x_{i_{\ell}}^{\alpha_{\ell}}$

for all compositions $\alpha$ satisfying $\ell(\alpha)\leq\left|I\right|$.
When $I$ is infinite, this means that the $M_{\alpha}$ for all compositions
$\alpha$ form a basis of $\operatorname{QSym}(\\{x_{i}\\}_{i\in I})$.

Note that $\operatorname{QSym}(\\{x_{i}\\}_{i\in I})=\bigoplus_{n\geq
0}\operatorname{QSym}_{n}(\\{x_{i}\\}_{i\in I})$ is a graded
$\mathbf{k}$-module of finite type, where
$\operatorname{QSym}_{n}(\\{x_{i}\\}_{i\in I})$ is the $\mathbf{k}$-submodule
of quasisymmetric functions which are homogeneous of degree $n$. Letting
$\operatorname{Comp}$ denote the set of all compositions $\alpha$, and
$\operatorname{Comp}_{n}$ the compositions $\alpha$ of $n$ (that is,
compositions whose parts sum to $n$), the subset
$\\{M_{\alpha}\\}_{\alpha\in\operatorname{Comp}_{n};\
\ell(\alpha)\leq\left|I\right|}$ gives a $\mathbf{k}$-basis for
$\operatorname{QSym}_{n}(\\{x_{i}\\}_{i\in I})$.

######  Example 5.2.

Taking the variable set ${\mathbf{x}}=(x_{1}<x_{2}<\cdots)$ to define
$\operatorname{QSym}({\mathbf{x}})$, for $n=0,1,2,3$, one has these basis
elements in $\operatorname{QSym}_{n}({\mathbf{x}})$:

$\begin{array}[]{rll}M_{()}=M_{\varnothing}&=1&\\\ &&\\\
M_{(1)}&=x_{1}+x_{2}+x_{3}+\cdots&=m_{(1)}=s_{(1)}=e_{1}=h_{1}=p_{1}\\\ &&\\\
M_{(2)}&=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+\cdots&=m_{(2)}=p_{2}\\\
M_{(1,1)}&=x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}+\cdots&=m_{(1,1)}=e_{2}\\\ &&\\\
M_{(3)}&=x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+\cdots&=m_{(3)}=p_{3}\\\
M_{(2,1)}&=x_{1}^{2}x_{2}+x_{1}^{2}x_{3}+x_{2}^{2}x_{3}+\cdots&\\\
M_{(1,2)}&=x_{1}x_{2}^{2}+x_{1}x_{3}^{2}+x_{2}x_{3}^{2}+\cdots&\\\
M_{(1,1,1)}&=x_{1}x_{2}x_{3}+x_{1}x_{2}x_{4}+x_{1}x_{3}x_{4}+\cdots&=m_{(1,1,1)}=e_{3}\end{array}$

It is not obvious that $\operatorname{QSym}({\mathbf{x}})$ is a subalgebra of
$R({\mathbf{x}})$, but we will show this momentarily. For example,

$\displaystyle M_{(a)}M_{(b,c)}$
$\displaystyle=(x_{1}^{a}+x_{2}^{a}+x_{3}^{a}+\cdots)(x_{1}^{b}x_{2}^{c}+x_{1}^{b}x_{3}^{c}+x_{2}^{b}x_{3}^{c}+\cdots)$
$\displaystyle=x_{1}^{a+b}x_{2}^{c}+\cdots+x_{1}^{b}x_{3}^{a+c}+\cdots+x_{1}^{a}x_{2}^{b}x_{3}^{c}+\cdots+x_{1}^{b}x_{2}^{a}x_{3}^{c}+\cdots+x_{1}^{b}x_{2}^{c}x_{3}^{a}+\cdots$
$\displaystyle=M_{(a+b,c)}+M_{(b,a+c)}+M_{(a,b,c)}+M_{(b,a,c)}+M_{(b,c,a)}$

######  Proposition 5.3.

For any infinite totally ordered set $I$, one has that
$\operatorname{QSym}(\\{x_{i}\\}_{i\in I})$ is a $\mathbf{k}$-subalgebra of
$R(\\{x_{i}\\}_{i\in I})$, with multiplication in the $\\{M_{\alpha}\\}$-basis
as follows: Fix three pairwise disjoint chain posets
$(i_{1}<\cdots<i_{\ell})$, $(j_{1}<\cdots<j_{m})$ and $(k_{1}<k_{2}<\cdots)$.
Now, if
$\alpha=(\alpha_{1},\ldots,\alpha_{\ell}),\beta=(\beta_{1},\ldots,\beta_{m})$
then

(5.1) $M_{\alpha}M_{\beta}=\sum_{f}M_{\operatorname{wt}f}$

in which the sum is over all $p\in{\mathbb{N}}$ and all maps $f$ from the
disjoint union of two chains to a chain

(5.2)
$(i_{1}<\cdots<i_{\ell})\sqcup(j_{1}<\cdots<j_{m})\overset{f}{\longrightarrow}(k_{1}<\cdots<k_{p})$

which are both surjective and strictly order-preserving ($x<y$ implies
$f(x)<f(y)$), and where the composition
$\operatorname{wt}(f):=(\operatorname{wt}_{1}(f),\ldots,\operatorname{wt}_{p}(f))$
is defined by $\operatorname{wt}_{s}(f):=\sum_{i_{u}\in
f^{-1}(k_{s})}\alpha_{u}+\sum_{j_{v}\in f^{-1}(k_{s})}\beta_{v}.$

######  Example 5.4.

For this example, set $\alpha=\left(2,1\right)$ and
$\beta=\left(3,4,2\right)$. Let us compute $M_{\alpha}M_{\beta}$ using (5.1).
Indeed, the length of $\alpha$ is $\ell=2$, and the length of $\beta$ is
$m=3$, so the sum on the right hand side of (5.1) is a sum over all
$p\in{\mathbb{N}}$ and all surjective strictly order-preserving maps $f$ from
the disjoint union $(i_{1}<i_{2})\sqcup(j_{1}<j_{2}<j_{3})$ of two chains to
the chain $(k_{1}<k_{2}<\cdots<k_{p})$. Such maps can exist only when $p\leq
5$ (due to having to be surjective) and only for $p\geq 3$ (since, being
strictly order-preserving, they have to be injective when restricted to
$(j_{1}<j_{2}<j_{3})$). Hence, enumerating them is a finite problem. The
reader can check that the value obtained fo $M_{\alpha}M_{\beta}$ is

$\displaystyle
M_{\left(2,1,3,4,2\right)}+M_{\left(2,3,1,4,2\right)}+M_{\left(2,3,4,1,2\right)}+M_{\left(2,3,4,2,1\right)}+M_{\left(3,2,1,4,2\right)}$
$\displaystyle+M_{\left(3,2,4,1,2\right)}+M_{\left(3,2,4,2,1\right)}+M_{\left(3,4,2,1,2\right)}+M_{\left(3,4,2,2,1\right)}+M_{\left(3,4,2,2,1\right)}$
$\displaystyle+M_{\left(2,3,4,3\right)}+M_{\left(2,3,5,2\right)}+M_{\left(2,4,4,2\right)}+M_{\left(3,2,4,3\right)}+M_{\left(3,2,5,2\right)}+M_{\left(3,4,2,3\right)}$
$\displaystyle+M_{\left(3,4,4,1\right)}+M_{\left(3,6,1,2\right)}+M_{\left(3,6,2,1\right)}+M_{\left(5,1,4,2\right)}+M_{\left(5,4,1,2\right)}+M_{\left(5,4,2,1\right)}$
$\displaystyle+M_{\left(5,4,3\right)}+M_{\left(5,5,2\right)}+M_{\left(3,6,3\right)}.$

Here, we have listed the addends corresponding to $p=5$ on the first two rows,
the addends corresponding to $p=4$ on the next two rows, and those
corresponding to $p=3$ on the third row. The reader might notice that the
first two rows (i.e., the addends with $p=5$) are basically a list of shuffles
of $\alpha$ and $\beta$: In general, the maps (5.2) for $p=\ell+m$ are in
bijection with the elements of $\operatorname{Sh}_{\ell,m}$ 197197197The
bijection takes a map $f$ to the inverse of the permutation
$\sigma\in\mathfrak{S}_{p}$ which sends every
$x\in\left\\{1,2,\ldots,\ell\right\\}$ to the index $y$ satisfying
$f\left(i_{x}\right)=k_{y}$, and sends every
$x\in\left\\{\ell+1,\ell+2,\ldots,\ell+m\right\\}$ to the index $y$ satisfying
$f\left(j_{x-\ell}\right)=k_{y}$., and the corresponding compositions
$\operatorname{wt}f$ are the shuffles of $\alpha$ and $\beta$. Therefore the
name “overlapping shuffle product”.

###### Proof of Proposition 5.3.

Formula (5.1) comes from considering a typical product of two monomials in the
expansion of $M_{\alpha}M_{\beta}$:

$\left(x_{i_{1}}^{\alpha_{1}}\cdots
x_{i_{\ell}}^{\alpha_{\ell}}\right)\left(x_{j_{1}}^{\beta_{1}}\cdots
x_{j_{m}}^{\beta_{m}}\right)=x_{k_{1}}^{\gamma_{1}}\cdots
x_{k_{p}}^{\gamma_{p}}$

for subscript sequences $i_{1}<\cdots<i_{\ell}$ and $j_{1}<\cdots<j_{m}$ and
$k_{1}<\cdots<k_{p}$ with

$\\{k_{1},\ldots,k_{p}\\}=\\{i_{1},\ldots,i_{\ell}\\}\cup\\{j_{1},\ldots,j_{m}\\}.$

Thinking of $\\{i_{s}\\}$ and $\\{j_{t}\\}$ as disjoint sets, multiplication
gives a surjective map $f$ as in (5.2), with
$\gamma_{s}=\operatorname{wt}_{s}(f)$. ∎

The multiplication rule (5.1) shows that the $\mathbf{k}$-algebra
$\operatorname{QSym}(\\{x_{i}\\}_{i\in I})$ does not depend much on $I$, as
long as $I$ is infinite. More precisely, all such $\mathbf{k}$-algebras are
mutually isomorphic. We can use this to define a $\mathbf{k}$-algebra of
quasisymmetric functions without any reference to $I$:

######  Definition 5.5.

Let $\operatorname{QSym}$ be the $\mathbf{k}$-algebra defined as having
$\mathbf{k}$-basis $\\{M_{\alpha}\\}_{\alpha\in\operatorname{Comp}}$ and with
multiplication defined $\mathbf{k}$-linearly by (5.1). This is called the
$\mathbf{k}$-algebra of quasisymmetric functions. We write
$\operatorname{QSym}_{\mathbf{k}}$ instead of $\operatorname{QSym}$ to stress
the choice of base ring $\mathbf{k}$.

The $\mathbf{k}$-algebra $\operatorname{QSym}$ is graded, and its $n$-th
graded component $\operatorname{QSym}_{n}$ has $\mathbf{k}$-basis
$\\{M_{\alpha}\\}_{\alpha\in\operatorname{Comp}_{n}}$.

For every infinite totally ordered set $I$, the $\mathbf{k}$-algebra
$\operatorname{QSym}$ is isomorphic to the $\mathbf{k}$-algebra
$\operatorname{QSym}(\\{x_{i}\\}_{i\in I})$. The isomorphism sends
$M_{\alpha}\longmapsto M_{\alpha}(\\{x_{i}\\}_{i\in I})$.

In particular, we obtain the isomorphism
$\operatorname{QSym}\cong\operatorname{QSym}\left({\mathbf{x}}\right)$ for
${\mathbf{x}}$ being the infinite chain
$\left(x_{1}<x_{2}<x_{3}<\cdots\right)$. We will identify
$\operatorname{QSym}$ with $\operatorname{QSym}\left({\mathbf{x}}\right)$
along this isomorphism. This allows us to regard quasisymmetric functions
either as power series in a specific set of variables (“alphabet”), or as
formal linear combinations of $M_{\alpha}$’s, whatever is more convenient.

For any infinite alphabet $\left\\{x_{i}\right\\}_{i\in I}$ and any
$f\in\operatorname{QSym}$, we denote by $f\left(\left\\{x_{i}\right\\}_{i\in
I}\right)$ the image of $f$ under the algebra isomorphism
$\operatorname{QSym}\to\operatorname{QSym}\left(\left\\{x_{i}\right\\}_{i\in
I}\right)$ defined in Definition 5.5.

The comultiplication of $\operatorname{QSym}$ will extend the one that we
defined for $\Lambda$, but we need to take care about the order of the
variables this time. We consider the linear order from (2.22) on two sets of
variables
$({\mathbf{x}},{\mathbf{y}})=(x_{1}<x_{2}<\ldots<y_{1}<y_{2}<\ldots),$ and we
embed the $\mathbf{k}$-algebra
$\operatorname{QSym}({\mathbf{x}})\otimes\operatorname{QSym}({\mathbf{y}})$
into the $\mathbf{k}$-algebra $R({\mathbf{x}},{\mathbf{y}})$ by identifying
every $f\otimes
g\in\operatorname{QSym}({\mathbf{x}})\otimes\operatorname{QSym}({\mathbf{y}})$
with $fg\in R({\mathbf{x}},{\mathbf{y}})$ (this embedding is indeed
injective198198198This is because it sends the basis elements
$M_{\beta}({\mathbf{x}})\otimes M_{\gamma}({\mathbf{y}})$ of the former
$\mathbf{k}$-algebra to the linearly independent power series
$M_{\beta}({\mathbf{x}})M_{\gamma}({\mathbf{y}})$.). It can then be seen that

$\operatorname{QSym}({\mathbf{x}},{\mathbf{y}})\subset\operatorname{QSym}({\mathbf{x}})\otimes\operatorname{QSym}({\mathbf{y}})$

(where the right hand side is viewed as $\mathbf{k}$-subalgebra of
$R({\mathbf{x}},{\mathbf{y}})$ via said embedding)199199199This is not
completely obvious, but can be easily checked by verifying that
$M_{\alpha}({\mathbf{x}},{\mathbf{y}})=\sum_{\begin{subarray}{c}(\beta,\gamma):\\\
\beta\cdot\gamma=\alpha\end{subarray}}M_{\beta}({\mathbf{x}})\otimes
M_{\gamma}({\mathbf{y}})$ for every composition $\alpha$ (see the proof of
Proposition 5.7 for why this holds)., so that one can define
$\operatorname{QSym}\overset{\Delta}{\longrightarrow}\operatorname{QSym}\otimes\operatorname{QSym}$
as the composite of the maps in the bottom row here:

(5.3)
$\begin{array}[]{rccccl}&&R({\mathbf{x}},{\mathbf{y}})&=&R({\mathbf{x}},{\mathbf{y}})\\\
&&\cup&&\cup\\\
\operatorname{QSym}&\cong&\operatorname{QSym}({\mathbf{x}},{\mathbf{y}})&\hookrightarrow&\operatorname{QSym}({\mathbf{x}})\otimes\operatorname{QSym}({\mathbf{y}})&\cong\operatorname{QSym}\otimes\operatorname{QSym}\\\
f&\longmapsto&f({\mathbf{x}},{\mathbf{y}})=f(x_{1},x_{2},\ldots,y_{1},y_{2},\ldots)&&\end{array}$

(Recall that $f({\mathbf{x}},{\mathbf{y}})$ is formally defined as the image
of $f$ under the algebra isomorphism
$\operatorname{QSym}\to\operatorname{QSym}({\mathbf{x}},{\mathbf{y}})$ defined
in Definition 5.5.)

######  Example 5.6.

For example,

$\displaystyle\Delta M_{(a,b,c)}$
$\displaystyle=M_{(a,b,c)}(x_{1},x_{2},\ldots,y_{1},y_{2},\ldots)$
$\displaystyle=x_{1}^{a}x_{2}^{b}x_{3}^{c}+x_{1}^{a}x_{2}^{b}x_{4}^{c}+\cdots$
$\displaystyle\quad+x_{1}^{a}x_{2}^{b}\cdot y_{1}^{c}+x_{1}^{a}x_{2}^{b}\cdot
y_{2}^{c}+\cdots$ $\displaystyle\quad+x_{1}^{a}\cdot
y_{1}^{b}y_{2}^{c}+x_{1}^{a}\cdot y_{1}^{b}y_{3}^{c}+\cdots$
$\displaystyle\quad+y_{1}^{a}y_{2}^{b}y_{3}^{c}+y_{1}^{a}y_{2}^{b}y_{4}^{c}+\cdots$
$\displaystyle=M_{(a,b,c)}({\mathbf{x}})+M_{(a,b)}({\mathbf{x}})M_{(c)}({\mathbf{y}})+M_{(a)}({\mathbf{x}})M_{(b,c)}({\mathbf{y}})+M_{(a,b,c)}({\mathbf{y}})$
$\displaystyle=M_{(a,b,c)}\otimes 1+M_{(a,b)}\otimes M_{(c)}+M_{(a)}\otimes
M_{(b,c)}+1\otimes M_{(a,b,c)}$

Defining the concatenation $\beta\cdot\gamma$ of two compositions
$\beta=(\beta_{1},\ldots,\beta_{r}),\gamma=(\gamma_{1},\ldots,\gamma_{s})$ to
be the composition
$(\beta_{1},\ldots,\beta_{r},\gamma_{1},\ldots,\gamma_{s})$, one has the
following description of the coproduct in the $\\{M_{\alpha}\\}$ basis.

######  Proposition 5.7.

For a composition $\alpha=(\alpha_{1},\ldots,\alpha_{\ell})$, one has

$\Delta M_{\alpha}=\sum_{k=0}^{\ell}M_{(\alpha_{1},\ldots,\alpha_{k})}\otimes
M_{(\alpha_{k+1},\ldots,\alpha_{\ell})}=\sum_{\begin{subarray}{c}(\beta,\gamma):\\\
\beta\cdot\gamma=\alpha\end{subarray}}M_{\beta}\otimes M_{\gamma}$

###### Proof.

This comes from expressing a monomial in $\Delta
M_{\alpha}=M_{\alpha}({\mathbf{x}},{\mathbf{y}})$ uniquely in the form
$x_{i_{1}}^{\alpha_{1}}\cdots x_{i_{k}}^{\alpha_{k}}\cdot
y_{j_{1}}^{\alpha_{k+1}}\cdots y_{j_{\ell-k}}^{\alpha_{\ell}}$ for some
$k\in\left\\{0,1,\ldots,n\right\\}$ and some subscripts $i_{1}<\cdots<i_{k}$
and $j_{1}<\cdots<j_{\ell-k}$. ∎

######  Proposition 5.8.

The quasisymmetric functions $\operatorname{QSym}$ form a graded connected
Hopf algebra of finite type, which is commutative, and contains the symmetric
functions $\Lambda$ as a Hopf subalgebra.

###### Proof.

To prove coassociativity of $\Delta$, we need to be slightly careful. It seems
reasonable to argue by $(\Delta\otimes\operatorname{id})\circ\Delta
f=f({\mathbf{x}},{\mathbf{y}},{\mathbf{z}})=(\operatorname{id}\otimes\Delta)\circ\Delta
f$ as in the case of $\Lambda$, but this would now require further
justification, as terms like $f({\mathbf{x}},{\mathbf{y}})$ and
$f({\mathbf{x}},{\mathbf{y}},{\mathbf{z}})$ are no longer directly defined as
evaluations of $f$ on some sequences (but rather are defined as images of $f$
under certain homomorphisms). However, it is very easy to see that $\Delta$ is
coassociative by checking
$(\Delta\otimes\operatorname{id})\circ\Delta=(\operatorname{id}\otimes\Delta)\circ\Delta$
on the $\left\\{M_{\alpha}\right\\}$ basis: Proposition 5.7 yields

$\displaystyle\left((\Delta\otimes\operatorname{id})\circ\Delta\right)M_{\alpha}$
$\displaystyle=\sum_{k=0}^{\ell}\Delta(M_{(\alpha_{1},\ldots,\alpha_{k})})\otimes
M_{(\alpha_{k+1},\ldots,\alpha_{\ell})}$
$\displaystyle=\sum_{k=0}^{\ell}\left(\sum_{i=0}^{k}M_{(\alpha_{1},\ldots,\alpha_{i})}\otimes
M_{(\alpha_{i+1},\ldots,\alpha_{k})}\right)\otimes
M_{(\alpha_{k+1},\ldots,\alpha_{\ell})}$
$\displaystyle=\sum_{k=0}^{\ell}\sum_{i=0}^{k}M_{(\alpha_{1},\ldots,\alpha_{i})}\otimes
M_{(\alpha_{i+1},\ldots,\alpha_{k})}\otimes
M_{(\alpha_{k+1},\ldots,\alpha_{\ell})}$

and the same expression for
$\left((\operatorname{id}\otimes\Delta)\circ\Delta\right)M_{\alpha}$.

The coproduct $\Delta$ of $\operatorname{QSym}$ is an algebra morphism because
it is defined as a composite of algebra morphisms in the bottom row of (5.3).
To prove that the restriction of $\Delta$ to the subring $\Lambda$ of
$\operatorname{QSym}$ is the comultiplication of $\Lambda$, it thus is enough
to check that it sends the elementary symmetric function $e_{n}$ to
$\sum_{i=0}^{n}e_{i}\otimes e_{n-i}$ for every $n\in{\mathbb{N}}$. This again
follows from Proposition 5.7, since $e_{n}=M_{(1,1,\ldots,1)}$ (with $n$ times
$1$).

The counit is as usual for a graded connected coalgebra, and just as in the
case of $\Lambda$, sends a quasisymmetric function $f({\mathbf{x}})$ to its
constant term $f(0,0,\ldots)$. This is an evaluation, and hence an algebra
map. Hence $\operatorname{QSym}$ forms a bialgebra, and as it is graded and
connected, also a Hopf algebra by Proposition 1.36. It is clearly of finite
type and contains $\Lambda$ as a Hopf subalgebra. ∎

We will identify the antipode in $\operatorname{QSym}$ shortly, but we first
deal with another slightly subtle issue. In addition to the counit evaluation
$\epsilon(f)=f(0,0,\ldots)$, starting in Section 7.1, we will want to
specialize elements in $\operatorname{QSym}({\mathbf{x}})$ by making other
variable substitutions, in which all but a finite list of variables are set to
zero. We justify this here.

######  Proposition 5.9.

Fix a totally ordered set $I$, a commutative $\mathbf{k}$-algebra $A$, a
finite list of variables $x_{i_{1}},\ldots,x_{i_{m}}$, say with
$i_{1}<\ldots<i_{m}$ in $I$, and an ordered list of elements
$(a_{1},\ldots,a_{m})\in A^{m}$.

Then there is a well-defined evaluation homomorphism

$\begin{array}[]{rcl}\operatorname{QSym}(\\{x_{i}\\}_{i\in
I})&\longrightarrow&A\\\
f&\longmapsto&\left[f\right]_{\begin{subarray}{c}x_{i_{1}}=a_{1},\ldots,x_{i_{m}}=a_{m}\\\
x_{j}=0\text{ for
}j\not\in\\{i_{1},\ldots,i_{m}\\}\end{subarray}}.\end{array}$

Furthermore, this homomorphism depends only upon the list
$(a_{1},\ldots,a_{m})$, as it coincides with the following:

$\begin{array}[]{rccl}\operatorname{QSym}(\\{x_{i}\\}_{i\in
I})&\cong\operatorname{QSym}(x_{1},x_{2},\ldots)&\longrightarrow&A\\\
&f(x_{1},x_{2},\ldots)&\longmapsto&f(a_{1},\ldots,a_{m},0,0\ldots).\end{array}$

(This latter statement is stated for the case when $I$ is infinite; otherwise,
read “$x_{1},x_{2},\ldots,x_{\left|I\right|}$” for “$x_{1},x_{2},\ldots$”, and
interpret $(a_{1},\ldots,a_{m},0,0\ldots)$ as an $\left|I\right|$-tuple.)

###### Proof.

One already can make sense of evaluating
$x_{i_{1}}=a_{1},\ldots,x_{i_{m}}=a_{m}$ and $x_{j}=0$ for
$j\not\in\\{i_{1},\ldots,i_{m}\\}$ in the ambient ring $R(\\{x_{i}\\}_{i\in
I})$ containing $\operatorname{QSym}(\\{x_{i}\\}_{i\in I})$, since a power
series $f$ of bounded degree will have finitely many monomials that only
involve the variables $x_{i_{1}},\ldots,x_{i_{m}}$. The last assertion follows
from quasisymmetry of $f$, and is perhaps checked most easily when
$f=M_{\alpha}(\\{x_{i}\\}_{i\in I})$ for some $\alpha$. ∎

The antipode in $\operatorname{QSym}$ has a reasonably simple expression in
the $\\{M_{\alpha}\\}$ basis, but requiring a definition.

######  Definition 5.10.

For $\alpha,\beta$ in $\operatorname{Comp}_{n}$, say that $\alpha$ refines
$\beta$ or $\beta$ coarsens $\alpha$ if, informally, one can obtain $\beta$
from $\alpha$ by combining some of its adjacent parts. Alternatively, one has
a bijection $\operatorname{Comp}_{n}\rightarrow 2^{[n-1]}$ where
$[n-1]:=\\{1,2,\ldots,n-1\\}$ which sends
$\alpha=(\alpha_{1},\ldots,\alpha_{\ell})$ having length $\ell(\alpha)=\ell$
to its subset of partial sums

$D(\alpha):=\left\\{\alpha_{1},\alpha_{1}+\alpha_{2},\ldots,\alpha_{1}+\cdots+\alpha_{\ell-1}\right\\},$

and this sends the refinement ordering to the inclusion ordering on the
Boolean algebra $2^{[n-1]}$ (to be more precise: a composition $\alpha$
refines a composition $\beta$ if and only if $D(\alpha)\supset D(\beta)$).
There is also a bijection sending $\alpha$ to its ribbon diagram: the skew
diagram $\lambda/\mu$ having rows of sizes $(\alpha_{1},\ldots,\alpha_{\ell})$
read from bottom to top with exactly one column of overlap between adjacent
rows. These bijections and the refinement partial order are illustrated here
for $n=4$:

$\textstyle{\\{1,2,3\\}}$$\textstyle{\\{1,2\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{1,3\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{2,3\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{1\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{2\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{3\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\varnothing\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1,1,1,1)}$$\textstyle{(1,1,2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1,2,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(2,1,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1,3)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(2,2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(3,1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(4)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\square$ $\square$ $\square$ $\square$ $\square\square$
$\square\phantom{\square}$ $\square\phantom{\square}$
$\phantom{\square}\square$ $\square\square$ $\square\phantom{\square}$
$\phantom{\square}\square$ $\phantom{\square}\square$ $\square\square$
$\square\square\square$ $\square\phantom{\square}\phantom{\square}$
$\phantom{\square}\square\square$ $\square\square\phantom{\square}$
$\phantom{\square}\phantom{\square}\square$ $\square\square\square$
$\square\square\square\square$

Given $\alpha=(\alpha_{1},\ldots,\alpha_{\ell})$, its reverse composition is
$\operatorname{rev}(\alpha)=(\alpha_{\ell},\alpha_{\ell-1},\ldots,\alpha_{2},\alpha_{1})$.
Note that $\alpha\mapsto\operatorname{rev}(\alpha)$ is a poset automorphism
for the refinement ordering.

######  Theorem 5.11.

For any composition $\alpha$ in $\operatorname{Comp}$,

$S(M_{\alpha})=(-1)^{\ell(\alpha)}\sum_{\begin{subarray}{c}\gamma\in\operatorname{Comp}:\\\
\gamma\text{ coarsens }\operatorname{rev}(\alpha)\end{subarray}}M_{\gamma}$

For example,

$S(M_{(a,b,c)})=-\left(M_{(c,b,a)}+M_{(b+c,a)}+M_{(c,a+b)}+M_{(a+b+c)}\right)$

###### Proof.

We give Ehrenborg’s proof200200200A different proof was given by Malvenuto and
Reutenauer [118, Cor. 2.3], and is sketched in Remark 5.36 below. [51, Prop.
3.4] via induction on $\ell=\ell(\alpha)$. One has easy base cases when
$\ell(\alpha)=0$, where
$S(M_{\varnothing})=S(1)=1=(-1)^{0}M_{\operatorname{rev}(\varnothing)},$ and
when $\ell(\alpha)=1$, where $M_{(n)}$ is primitive by Proposition 5.7, so
Proposition 1.37 shows
$S(M_{(n)})=-M_{(n)}=(-1)^{1}M_{\operatorname{rev}((n))}.$

For the inductive step, apply the inductive definition of $S$ from the proof
of Proposition 1.36:

$\displaystyle S(M_{(\alpha_{1},\ldots,\alpha_{\ell})})$
$\displaystyle=-\sum_{i=0}^{\ell-1}S(M_{(\alpha_{1},\ldots,\alpha_{i})})M_{(\alpha_{i+1},\ldots,\alpha_{\ell})}$
$\displaystyle=\sum_{i=0}^{\ell-1}\sum_{\begin{subarray}{c}\beta\text{
coarsening }\\\
(\alpha_{i},\alpha_{i-1},\ldots,\alpha_{1})\end{subarray}}(-1)^{i+1}M_{\beta}M_{(\alpha_{i+1},\ldots,\alpha_{\ell})}$

The idea will be to cancel terms of opposite sign that appear in the
expansions of the products $M_{\beta}M_{(\alpha_{i+1},\ldots,\alpha_{\ell})}$.
Note that each composition $\beta$ appearing above has first part $\beta_{1}$
of the form $\alpha_{i}+\alpha_{i-1}+\cdots+\alpha_{h}$ for some $h\leq i$
(unless $\beta=\varnothing$), and hence each term $M_{\gamma}$ in the
expansion of the product $M_{\beta}M_{(\alpha_{i+1},\ldots,\alpha_{\ell})}$
has $\gamma_{1}$ (that is, the first entry of $\gamma$) a sum that can take
one of these three forms:

  1. $\bullet$

$\alpha_{i}+\alpha_{i-1}+\cdots+\alpha_{h}$,

  2. $\bullet$

$\alpha_{i+1}+(\alpha_{i}+\alpha_{i-1}+\cdots+\alpha_{h})$,

  3. $\bullet$

$\alpha_{i+1}$.

Say that the type of $\gamma$ is $i$ in the first case, and $i+1$ in the
second two cases201201201We imagine that we label the terms obtained by
expanding $M_{\beta}M_{(\alpha_{i+1},\ldots,\alpha_{\ell})}$ by distinct
labels, so that each term knows how exactly it was created (i.e., which $i$,
which $\beta$ and which map $f$ as in (5.2) gave rise to it). Strictly
speaking, it is these triples $\left(i,\beta,f\right)$ that we should be
assigning types to, not terms.; in other words, the type is the largest
subscript $k$ on a part $\alpha_{k}$ which was combined in the sum
$\gamma_{1}$. It is not hard to see that a given $\gamma$ for which the type
$k$ is strictly smaller than $\ell$ arises from exactly two pairs
$(\beta,\gamma),(\beta^{\prime},\gamma)$, having opposite signs $(-1)^{k}$ and
$(-1)^{k+1}$ in the above sum202202202Strictly speaking, this means that we
have an involution on the set of our $\left(i,\beta,f\right)$ triples having
type smaller than $\ell$, and this involution switches the sign of
$(-1)^{i}M_{\operatorname{wt}f}$.. For example, if
$\alpha=(\alpha_{1},\ldots,\alpha_{8})$, then the composition
$\gamma=(\alpha_{6}+\alpha_{5}+\alpha_{4},\alpha_{3},\alpha_{7},\alpha_{8}+\alpha_{2}+\alpha_{1})$
of type $6$ can arise from either of

$\displaystyle\beta$
$\displaystyle=(\alpha_{6}+\alpha_{5}+\alpha_{4},\alpha_{3},\alpha_{2}+\alpha_{1})\text{
with }i=6\text{ and sign }(-1)^{7}$ $\displaystyle\beta^{\prime}$
$\displaystyle=(\alpha_{5}+\alpha_{4},\alpha_{3},\alpha_{2}+\alpha_{1})\text{
with }i=5\text{ and sign }(-1)^{6}.$

Similarly,
$\gamma=(\alpha_{6},\alpha_{5}+\alpha_{4},\alpha_{3},\alpha_{7},\alpha_{8}+\alpha_{2}+\alpha_{1})$
can arise from either of

$\displaystyle\beta$
$\displaystyle=(\alpha_{6},\alpha_{5}+\alpha_{4},\alpha_{3},\alpha_{2}+\alpha_{1})\text{
with }i=6\text{ and sign }(-1)^{7}$ $\displaystyle\beta^{\prime}$
$\displaystyle=(\alpha_{5}+\alpha_{4},\alpha_{3},\alpha_{2}+\alpha_{1})\text{
with }i=5\text{ and sign }(-1)^{6}.$

Thus one can cancel almost all the terms, excepting those with $\gamma$ of
type $\ell$ among the terms $M_{\gamma}$ in the expansion of the last
($i=\ell-1$) summand $M_{\beta}M_{(\alpha_{\ell})}$. A bit of thought shows
that these are the $\gamma$ coarsening $\operatorname{rev}(\alpha)$, and all
have sign $(-1)^{\ell}$. ∎

###  5.2. The fundamental basis and $P$-partitions

There is a second important basis for $\operatorname{QSym}$ which arose
originally in Stanley’s $P$-partition theory [161].203203203See [64] for a
history of $P$-partitions; our notations, however, strongly differ from those
in [64].

######  Definition 5.12.

A labelled poset will here mean a partially ordered set $P$ whose underlying
set is some finite subset of the integers. A $P$-partition is a function
$P\overset{f}{\rightarrow}\\{1,2,\ldots\\}$ with the property that

  1. $\bullet$

$i<_{P}j$ and $i<_{\mathbb{Z}}j$ implies $f(i)\leq f(j)$, and

  2. $\bullet$

$i<_{P}j$ and $i>_{\mathbb{Z}}j$ implies $f(i)<f(j)$.

Denote by ${\mathcal{A}}(P)$ the set of all $P$-partitions $f$, and let
$F_{P}({\mathbf{x}}):=\sum_{f\in{\mathcal{A}}(P)}{\mathbf{x}}_{f}$ where
${\mathbf{x}}_{f}:=\prod_{i\in P}x_{f(i)}$. This $F_{P}({\mathbf{x}})$ is an
element of
$\mathbf{k}\left[\left[{\mathbf{x}}\right]\right]:=\mathbf{k}\left[\left[x_{1},x_{2},\ldots\right]\right]$.

######  Example 5.13.

Depicted is a labelled poset $P$, along with the relations among the four
values $f=(f(1),f(2),f(3),f(4))$ that define its $P$-partitions $f$:

$\textstyle{2}$$\textstyle{4}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{f(2)}$$\textstyle{f(4)}$$\textstyle{f(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\leq}$$\textstyle{f(3)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\leq}$$\scriptstyle{<}$

######  Remark 5.14.

Stanley’s treatment of $P$-partitions in [164, §3.15 and §7.19] uses a
language different from ours. First, Stanley works not with labelled posets
$P$, but with pairs $\left(P,\omega\right)$ of a poset $P$ and a bijective
labelling $\omega:P\to[n]$. Thus, the relation $<_{\mathbb{Z}}$ is not given
on $P$ a priori, but has to be pulled back from $[n]$ using $\omega$ (and it
depends on $\omega$, whence Stanley speaks of
“$\left(P,\omega\right)$-partitions”). Furthermore, Stanley uses the notations
$F_{P}$ and $F_{P,\omega}$ for something different from what we denote by
$F_{P}$, whereas what we call $F_{P}$ is dubbed $K_{P,\omega}$ in [164,
§7.19].

The so-called fundamental quasisymmetric functions are an important special
case of the $F_{P}({\mathbf{x}})$. We shall first define them directly and
then see how they are obtained as $P$-partition enumerators
$F_{P}({\mathbf{x}})$ for some special labelled posets $P$.

######  Definition 5.15.

Let $n\in{\mathbb{N}}$ and $\alpha\in\operatorname{Comp}_{n}$. We define the
fundamental quasisymmetric function
$L_{\alpha}=L_{\alpha}({\mathbf{x}})\in\operatorname{QSym}$ by

(5.4)
$L_{\alpha}:=\sum_{\begin{subarray}{c}\beta\in\operatorname{Comp}_{n}:\\\
\beta\text{ refines }\alpha\end{subarray}}M_{\beta}.$

######  Example 5.16.

The extreme cases for $\alpha$ in $\operatorname{Comp}_{n}$ give
quasisymmetric functions $L_{\alpha}$ which are symmetric:

$\displaystyle L_{(1^{n})}$ $\displaystyle=M_{(1^{n})}=e_{n},$ $\displaystyle
L_{(n)}$
$\displaystyle=\sum_{\alpha\in\operatorname{Comp}_{n}}M_{\alpha}=h_{n}$

One sees that the $\\{L_{\alpha}\\}_{\alpha\in\operatorname{Comp}}$ are a
$\mathbf{k}$-basis for $\operatorname{QSym}$, as inclusion-exclusion applied
to (5.4) gives

$M_{\alpha}=\sum_{\begin{subarray}{c}\beta\in\operatorname{Comp}_{n}:\\\
\beta\text{ refines
}\alpha\end{subarray}}(-1)^{\ell(\beta)-\ell(\alpha)}L_{\beta}.$

######  Proposition 5.17.

Let $n\in{\mathbb{N}}$. Let $\alpha$ be a composition of $n$. Let $I$ be an
infinite totally ordered set. Then,

$L_{\alpha}\left(\left\\{x_{i}\right\\}_{i\in
I}\right)=\sum_{\begin{subarray}{c}i_{1}\leq i_{2}\leq\cdots\leq i_{n}\text{
in }I;\\\ i_{j}<i_{j+1}\text{ if }j\in
D\left(\alpha\right)\end{subarray}}x_{i_{1}}x_{i_{2}}\cdots x_{i_{n}},$

where $L_{\alpha}\left(\left\\{x_{i}\right\\}_{i\in I}\right)$ is defined as
the image of $L_{\alpha}$ under the isomorphism
$\operatorname{QSym}\to\operatorname{QSym}\left(\left\\{x_{i}\right\\}_{i\in
I}\right)$ obtained in Definition 5.5. In particular, for the standard
(totally ordered) variable set ${\mathbf{x}}=\left(x_{1}<x_{2}<\ldots\right)$,
we obtain

(5.5)
$L_{\alpha}=L_{\alpha}\left({\mathbf{x}}\right)=\sum_{\begin{subarray}{c}(1\leq)i_{1}\leq
i_{2}\leq\cdots\leq i_{n};\\\ i_{j}<i_{j+1}\text{ if }j\in
D\left(\alpha\right)\end{subarray}}x_{i_{1}}x_{i_{2}}\cdots x_{i_{n}}.$

###### Proof.

Every composition $\beta=\left(\beta_{1},\ldots,\beta_{\ell}\right)$ of $n$
satisfies

(5.6) $M_{\beta}\left(\\{x_{i}\\}_{i\in
I}\right)=\sum_{k_{1}<\cdots<k_{\ell}\text{ in }I}x_{k_{1}}^{\beta_{1}}\cdots
x_{k_{\ell}}^{\beta_{\ell}}=\sum_{\begin{subarray}{c}i_{1}\leq
i_{2}\leq\cdots\leq i_{n}\text{ in }I;\\\ i_{j}<i_{j+1}\text{ if and only if
}j\in D\left(\beta\right)\end{subarray}}x_{i_{1}}x_{i_{2}}\cdots x_{i_{n}}.$

Applying the ring homomorphism
$\operatorname{QSym}\to\operatorname{QSym}\left(\left\\{x_{i}\right\\}_{i\in
I}\right)$ to (5.4), we obtain

$\displaystyle L_{\alpha}\left(\left\\{x_{i}\right\\}_{i\in I}\right)$
$\displaystyle=\sum_{\begin{subarray}{c}\beta\in\operatorname{Comp}_{n}:\\\
\beta\text{ refines
}\alpha\end{subarray}}M_{\beta}\left(\left\\{x_{i}\right\\}_{i\in
I}\right)\overset{\eqref{Qsym-technical-lemma-Lxy-
proof}}{=}\sum_{\begin{subarray}{c}\beta\in\operatorname{Comp}_{n}:\\\
\beta\text{ refines
}\alpha\end{subarray}}\quad\sum_{\begin{subarray}{c}i_{1}\leq
i_{2}\leq\cdots\leq i_{n}\text{ in }I;\\\ i_{j}<i_{j+1}\text{ if and only if
}j\in D\left(\beta\right)\end{subarray}}x_{i_{1}}x_{i_{2}}\cdots x_{i_{n}}$
$\displaystyle=\sum_{\begin{subarray}{c}\beta\in\operatorname{Comp}_{n}:\\\
D(\alpha)\subset
D(\beta)\end{subarray}}\quad\sum_{\begin{subarray}{c}i_{1}\leq
i_{2}\leq\cdots\leq i_{n}\text{ in }I;\\\ i_{j}<i_{j+1}\text{ if and only if
}j\in D\left(\beta\right)\end{subarray}}x_{i_{1}}x_{i_{2}}\cdots x_{i_{n}}$
$\displaystyle=\sum_{\begin{subarray}{c}Z\subset[n-1]:\\\ D(\alpha)\subset
Z\end{subarray}}\quad\sum_{\begin{subarray}{c}i_{1}\leq i_{2}\leq\cdots\leq
i_{n}\text{ in }I;\\\ i_{j}<i_{j+1}\text{ if and only if }j\in
Z\end{subarray}}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{n}}=\sum_{\begin{subarray}{c}i_{1}\leq i_{2}\leq\cdots\leq i_{n}\text{
in }I;\\\ i_{j}<i_{j+1}\text{ if }j\in
D\left(\alpha\right)\end{subarray}}x_{i_{1}}x_{i_{2}}\cdots x_{i_{n}}.$

∎

######  Proposition 5.18.

When the labelled poset $P$ is a total or linear order
$w=(w_{1}<\ldots<w_{n})$, the generating function $F_{w}({\mathbf{x}})$
depends only upon the descent set

$\operatorname{Des}(w):=\\{i:w_{i}>_{{\mathbb{Z}}}w_{i+1}\\}\subset\\{1,2,\ldots,n-1\\}$

and its associated composition $\alpha$ in $\operatorname{Comp}_{n}$ having
partial sums $D(\alpha)=\operatorname{Des}(w)$: one has that
$F_{w}({\mathbf{x}})$ equals the fundamental quasisymmetric function
$L_{\alpha}$.

E.g., total order $w=35142$ has $\operatorname{Des}(w)=\\{2,4\\}$ and
composition $\alpha=(2,2,1)$, so

$\displaystyle F_{35142}({\mathbf{x}})$ $\displaystyle=\sum_{f(3)\leq
f(5)<f(1)\leq f(4)<f(2)}x_{f(3)}x_{f(5)}x_{f(1)}x_{f(4)}x_{f(2)}$
$\displaystyle=\sum_{i_{1}\leq i_{2}<i_{3}\leq
i_{4}<i_{5}}x_{i_{1}}x_{i_{2}}x_{i_{3}}x_{i_{4}}x_{i_{5}}$
$\displaystyle=L_{(2,2,1)}=M_{(2,2,1)}+M_{(2,1,1,1)}+M_{(1,1,2,1)}+M_{(1,1,1,1,1)}.$

###### Proof of Proposition 5.18..

Write $F_{w}({\mathbf{x}})$ as a sum of monomials $x_{f(w_{1})}\cdots
x_{f(w_{n})}$ over all $P$-partitions $f$. These $P$-partitions are exactly
the sequences $f(w_{1})\leq\cdots\leq f(w_{n})$ having strict inequalities
$f(w_{i})<f(w_{i+1})$ whenever $i$ is in $\operatorname{Des}(w)$. Hence,
comparison with (5.5) reveals that this sum equals $L_{\alpha}$. ∎

The next proposition ([164, Cor. 7.19.5], [112, Cor. 3.3.24]) is an algebraic
shadow of Stanley’s main lemma [164, Thm. 7.19.4] in $P$-partition theory. It
expands any $F_{P}({\mathbf{x}})$ in the $\\{L_{\alpha}\\}$ basis, as a sum
over the set $\mathcal{L}(P)$ of all linear extensions $w$ of $P$ 204204204Let
us explain what we mean by linear extensions and how we represent them. If
$\mathbf{P}$ is a finite poset, then a linear extension of $\mathbf{P}$
denotes a total order $w$ on the set $\mathbf{P}$ having the property that
$i<_{\mathbf{P}}j$ (for two elements $i$ and $j$ of $\mathbf{P}$) implies
$i<_{w}j$. (In other words, it is a linear order on the ground set
$\mathbf{P}$ which extends $\mathbf{P}$ as a poset; therefore the name.) We
identify such a total order $w$ with the list
$\left(\mathbf{p}_{1},\mathbf{p}_{2},\ldots,\mathbf{p}_{n}\right)$ containing
all elements of $\mathbf{P}$ in $w$-increasing order (that is,
$\mathbf{p}_{1}<_{w}\mathbf{p}_{2}<_{w}\cdots<_{w}\mathbf{p}_{n}$). (Stanley,
in [164, §3.5], defines linear extensions in a slightly different way: For
him, a linear extension of a finite poset $\mathbf{P}$ is an order-preserving
bijection from $\mathbf{P}$ to the subposet
$\left\\{1,2,\ldots,\left|\mathbf{P}\right|\right\\}$ of ${\mathbb{Z}}$. But
this is equivalent to our definition, since a bijection like this can be used
to transport the order relation of
$\left\\{1,2,\ldots,\left|\mathbf{P}\right|\right\\}$ back to $\mathbf{P}$,
thus resulting in a total order on $\mathbf{P}$ which is a linear extension of
$\mathbf{P}$ in our sense.) . E.g., the poset $P$ from Example 5.13 has
$\mathcal{L}(P)=\\{3124,3142,3412\\}$.

######  Theorem 5.19.

For any labelled poset $P$,

$F_{P}({\mathbf{x}})=\sum_{w\in\mathcal{L}(P)}F_{w}({\mathbf{x}}).$

###### Proof.

We give Gessel’s proof [63, Thm. 1], via induction on the number of pairs
$i,j$ which are incomparable in $P$. When this quantity is $0$, then $P$ is
itself a linear order $w$, so that $\mathcal{L}(P)=\\{w\\}$ and there is
nothing to prove.

In the inductive step, let $i,j$ be incomparable elements. Consider the two
posets $P_{i<j}$ and $P_{j<i}$ which are obtained from $P$ by adding in an
order relation between $i$ and $j$, and then taking the transitive closure; it
is not hard to see that these transitive closures cannot contain a cycle, so
that these really do define two posets. The result then follows by induction
applied to $P_{i<j},P_{j<i}$, once one notices that
$\mathcal{L}(P)=\mathcal{L}(P_{i<j})\sqcup\mathcal{L}(P_{j<i})$ since every
linear extension $w$ of $P$ either has $i$ before $j$ or vice-versa, and
${\mathcal{A}}(P)={\mathcal{A}}(P_{i<j})\sqcup{\mathcal{A}}(P_{j<i})$ since,
assuming that $i<_{\mathbb{Z}}j$ without loss of generality, every $f$ in
${\mathcal{A}}(P)$ either satisfies $f(i)\leq f(j)$ or $f(i)>f(j)$. ∎

######  Example 5.20.

To illustrate the induction in the above proof, consider the poset $P$ from
Example 5.13, having $\mathcal{L}(P)=\\{3124,3142,3412\\}$. Then choosing as
incomparable pair $(i,j)=(1,4)$, one has

$\textstyle{4}$$\textstyle{2}$$\textstyle{P_{i<j}=}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{f(4)}$$\textstyle{f(2)}$$\textstyle{f(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\leq}$$\scriptstyle{\leq}$$\textstyle{f(3)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{<}$
$\textstyle{\mathcal{L}(P_{i<j})=\\{3124,3142\\}}$
$\textstyle{2}$$\textstyle{P_{j<i}=}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{f(2)}$$\textstyle{f(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\leq}$$\textstyle{f(4)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{<}$$\textstyle{f(3)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\leq}$
$\textstyle{\mathcal{L}(P_{j<i})=\\{3412\\}}$

######  Exercise 5.21.

Give an alternative proof for Theorem 5.19.

[Hint: For every $f:P\rightarrow\left\\{1,2,3,\ldots\right\\}$, we can define
a binary relation $\prec_{f}$ on the set $P$ by letting $i\prec_{f}j$ hold if
and only if

$\left(f\left(i\right)<f\left(j\right)\text{ or
}\left(f\left(i\right)=f\left(j\right)\text{ and
}i<_{{\mathbb{Z}}}j\right)\right).$

Show that this binary relation $\prec_{f}$ is (the smaller relation of) a
total order. When $f$ is a $P$-partition, then endowing the set $P$ with this
total order yields a linear extension of $P$. Use this to show that the set
$\mathcal{A}\left(P\right)$ is the union of its pairwise disjoint subsets
$\mathcal{A}\left(w\right)$ with $w\in\mathcal{L}\left(P\right)$.]

Various other properties of the quasisymmetric functions
$F_{P}\left({\mathbf{x}}\right)$ are studied, e.g., in [123].

We next wish to describe the structure maps for the Hopf algebra
$\operatorname{QSym}$ in the basis $\\{L_{\alpha}\\}$ of fundamental
quasisymmetric functions. For this purpose, two more definitions are useful.

######  Definition 5.22.

Given two nonempty compositions
$\alpha=(\alpha_{1},\ldots,\alpha_{\ell}),\beta=(\beta_{1},\ldots,\beta_{m})$,
their near-concatenation is

$\alpha\odot\beta:=(\alpha_{1},\ldots,\alpha_{\ell-1},\alpha_{\ell}+\beta_{1},\beta_{2},\ldots,\beta_{m})$

For example, the figure below depicts for $\alpha=(1,3,3)$ (black squares) and
$\beta=(4,2)$ (white squares) the concatenation and near-concatenation as
ribbons:

$\displaystyle\alpha\cdot\beta$
$\displaystyle=\begin{matrix}&&&&&&&\square&\square\\\
&&&&\square&\square&\square&\square&\\\
&&\blacksquare&\blacksquare&\blacksquare&&&&\\\
\blacksquare&\blacksquare&\blacksquare&&&&&&\\\
\blacksquare&&&&&&&&\end{matrix}$ $\displaystyle\alpha\odot\beta$
$\displaystyle=\begin{matrix}&&&&&&&&\square&\square\\\
&&\blacksquare&\blacksquare&\blacksquare&\square&\square&\square&\square&\\\
\blacksquare&\blacksquare&\blacksquare&&&&&&&\\\
\blacksquare&&&&&&&&&\end{matrix}$

Lastly, given $\alpha$ in $\operatorname{Comp}_{n}$, let $\omega(\alpha)$ be
the unique composition in $\operatorname{Comp}_{n}$ whose partial sums
$D(\omega(\alpha))$ form the complementary set within $[n-1]$ to the partial
sums $D(\operatorname{rev}(\alpha))$; alternatively, one can check this means
that the ribbon for $\omega(\alpha)$ is obtained from that of $\alpha$ by
conjugation or transposing, that is, if $\alpha=\lambda/\mu$ then
$\omega(\alpha)=\lambda^{t}/\mu^{t}$. E.g. if $\alpha=(4,2,2)$ so that $n=8$,
then $\operatorname{rev}(\alpha)=(2,2,4)$ has
$D(\operatorname{rev}(\alpha))=\\{2,4\\}\subset[7]$, complementary to the set
$\\{1,3,5,6,7\\}$ which are the partial sums for
$\omega(\alpha)=(1,2,2,1,1,1)$, and the ribbon diagrams of
$\alpha,\omega(\alpha)$ are

$\alpha=\begin{matrix}&&&&\square&\square\\\ &&&\square&\square&\\\
\square&\square&\square&\square&&\\\ \end{matrix}\qquad\text{ and
}\qquad\omega(\alpha)=\begin{matrix}&&\square\\\ &&\square\\\ &&\square\\\
&\square&\square\\\ \square&\square&\\\ \square&&\\\ \end{matrix}$

######  Proposition 5.23.

The structure maps for the Hopf algebra $\operatorname{QSym}$ in the basis
$\\{L_{\alpha}\\}$ of fundamental quasisymmetric functions are as follows:

(5.7) $\displaystyle\Delta L_{\alpha}$
$\displaystyle=\sum_{\begin{subarray}{c}(\beta,\gamma):\\\
\beta\cdot\gamma=\alpha\text{ or
}\beta\odot\gamma=\alpha\end{subarray}}L_{\beta}\otimes L_{\gamma}$ (5.8)
$\displaystyle L_{\alpha}L_{\beta}$
$\displaystyle=\sum_{w\in{w_{\alpha}}\,\shuffle\,{w_{\beta}}}L_{\gamma(w)}$
(5.9) $\displaystyle S(L_{\alpha})$
$\displaystyle=(-1)^{|\alpha|}L_{\omega(\alpha)}.$

Here we are making use of the following notations in (5.8) (recall also
Definition 1.68):

  1. $\bullet$

A labelled linear order will mean a labelled poset $P$ whose order $<_{P}$ is
a total order. We will identify any labelled linear order $P$ with the word
(over the alphabet ${\mathbb{Z}}$) obtained by writing down the elements of
$P$ in increasing order (with respect to the total order $<_{P}$). This way,
every word (over the alphabet ${\mathbb{Z}}$) which has no two equal letters
becomes identified with a labelled linear order.

  2. $\bullet$

$w_{\alpha}$ is any labelled linear order with underlying set
$\left\\{1,2,\ldots,\left|\alpha\right|\right\\}$ such that
$\operatorname{Des}\left(w_{\alpha}\right)=D\left(\alpha\right)$.

  3. $\bullet$

$w_{\beta}$ is any labelled linear order with underlying set
$\left\\{\left|\alpha\right|+1,\left|\alpha\right|+2,\ldots,\left|\alpha\right|+\left|\beta\right|\right\\}$
such that $\operatorname{Des}\left(w_{\beta}\right)=D\left(\beta\right)$.

  4. $\bullet$

$\gamma(w)$ is the unique composition of
$\left|\alpha\right|+\left|\beta\right|$ with
$D(\gamma(w))=\operatorname{Des}(w)$.

At first glance the formula (5.7) for $\Delta L_{\alpha}$ might seem more
complicated than the formula of Proposition 5.7 for $\Delta M_{\alpha}$.
However, it is equally simple when viewed in terms of ribbon diagrams: it cuts
the ribbon diagram $\alpha$ into two smaller ribbons $\beta$ and $\gamma$, in
all $|\alpha|+1$ possible ways, via horizontal cuts
($\beta\cdot\gamma=\alpha$) or vertical cuts ($\beta\odot\gamma=\alpha$). For
example,

$\begin{array}[]{rcccccl}\Delta L_{(3,2)}&=1\otimes L_{(3,2)}&+L_{(1)}\otimes
L_{(2,2)}&+L_{(2)}\otimes L_{(1,2)}&+L_{(3)}\otimes L_{(2)}&+L_{(3,1)}\otimes
L_{(1)}&+L_{(3,2)}\otimes 1\\\ &\begin{matrix}&&\square&\square\\\
\underline{\square}&\square&\square&\end{matrix}&\begin{matrix}&&\square&\square\\\
\square&|\square&\square&\end{matrix}&\begin{matrix}&&\square&\square\\\
\square&\square&|\square&\end{matrix}&\begin{matrix}&&\underline{\square}&\square\\\
\square&\square&\square&\end{matrix}&\begin{matrix}&&\square&|\square\\\
\square&\square&\square&\end{matrix}&\begin{matrix}&&\square&\overline{\square}\\\
\square&\square&\square&\end{matrix}\end{array}$

######  Example 5.24.

To multiply $L_{(1,1)}L_{(2)}$, one could pick $w_{\alpha}=21$ and
$w_{\beta}=34$, and then

$\begin{array}[]{rcccccl}L_{(1,1)}L_{(2)}=\sum_{w\in{21}\,\shuffle\,{34}}L_{\gamma(w)}&=L_{\gamma(2134)}&+L_{\gamma(2314)}&+L_{\gamma(3214)}&+L_{\gamma(2341)}&+L_{\gamma(3241)}&+L_{\gamma(3421)}\\\
&=L_{(1,3)}&+L_{(2,2)}&+L_{(1,1,2)}&+L_{(3,1)}&+L_{(1,2,1)}&+L_{(2,1,1)}.\end{array}$

###### Proof of Proposition 5.23..

To prove formula (5.7) for $\alpha$ in $\operatorname{Comp}_{n}$, note that

(5.10) $\Delta
L_{\alpha}=L_{\alpha}({\mathbf{x}},{\mathbf{y}})=\sum_{k=0}^{n}\sum_{\begin{subarray}{c}1\leq
i_{1}\leq\cdots\leq i_{k},\\\ 1\leq i_{k+1}\leq\cdots\leq i_{n}:\\\
i_{r}<i_{r+1}\text{ for }r\in
D(\alpha)\setminus\left\\{k\right\\}\end{subarray}}x_{i_{1}}\cdots
x_{i_{k}}\cdot y_{i_{k+1}}\cdots y_{i_{n}}$

by Proposition 5.17 (where we identify
$\operatorname{QSym}\otimes\operatorname{QSym}$ with a $\mathbf{k}$-subalgebra
of $R\left({\mathbf{x}},{\mathbf{y}}\right)$ by means of the embedding
$\operatorname{QSym}\otimes\operatorname{QSym}\overset{\cong}{\rightarrow}\operatorname{QSym}\left({\mathbf{x}}\right)\otimes\operatorname{QSym}\left({\mathbf{y}}\right)\hookrightarrow
R\left({\mathbf{x}},{\mathbf{y}}\right)$ as in the definition of the
comultiplication on $\operatorname{QSym}$). One then realizes that the inner
sums corresponding to values of $k$ that lie (resp. do not lie) in
$D(\alpha)\cup\left\\{0,n\right\\}$ correspond to the terms
$L_{\beta}({\mathbf{x}})L_{\gamma}({\mathbf{y}})$ for pairs $(\beta,\gamma)$
in which $\beta\cdot\gamma=\alpha$ (resp. $\beta\odot\gamma=\alpha$).

For formula (5.8), let $P$ be the labelled poset which is the disjoint union
of linear orders $w_{\alpha},w_{\beta}$. Then

$L_{\alpha}L_{\beta}=F_{w_{\alpha}}({\mathbf{x}})F_{w_{\beta}}({\mathbf{x}})=F_{P}({\mathbf{x}})=\sum_{w\in\mathcal{L}(P)}F_{w}({\mathbf{x}})=\sum_{w\in{w_{\alpha}}\,\shuffle\,{w_{\beta}}}L_{\gamma(w)}$

where the first equality used Proposition 5.18, the second equality comes from
the definition of a $P$-partition, the third equality from Theorem 5.19, and
the fourth from the equality
$\mathcal{L}(P)={w_{\alpha}}\,\shuffle\,{w_{\beta}}$.

To prove formula (5.9), compute using Theorem 5.11 that

$S(L_{\alpha})=\sum_{\beta\text{ refining
}\alpha}S(M_{\beta})=\sum_{\begin{subarray}{c}(\beta,\gamma):\\\ \beta\text{
refines }\alpha,\\\ \gamma\text{ coarsens
}\operatorname{rev}(\beta)\end{subarray}}(-1)^{\ell(\beta)}M_{\gamma}=\sum_{\gamma}M_{\gamma}\sum_{\beta}(-1)^{\ell(\beta)}$

in which the last inner sum is over $\beta$ for which

$D(\beta)\supset D(\alpha)\cup D(\operatorname{rev}(\gamma)).$

The alternating signs make such inner sums vanish unless they have only the
single term where $D(\beta)=[n-1]$ (that is, $\beta=(1^{n})$). This happens
exactly when $D(\operatorname{rev}(\gamma))\cup D(\alpha)=[n-1]$ or
equivalently, when $D(\operatorname{rev}(\gamma))$ contains the complement of
$D(\alpha)$, that is, when $D(\gamma)$ contains the complement of
$D(\operatorname{rev}(\alpha))$, that is, when $\gamma$ refines
$\omega(\alpha)$. Thus

$S(L_{\alpha})=\sum_{\begin{subarray}{c}\gamma\in\operatorname{Comp}_{n}:\\\
\gamma\text{ refines
}\omega(\alpha)\end{subarray}}M_{\gamma}\cdot(-1)^{n}=(-1)^{|\alpha|}L_{\omega(\alpha)}.$

∎

The antipode formula (5.9) for $L_{\alpha}$ leads to a general interpretation
for the antipode of $\operatorname{QSym}$ acting on $P$-partition enumerators
$F_{P}({\mathbf{x}})$.

######  Definition 5.25.

Given a labelled poset $P$ on $\\{1,2,\ldots,n\\}$, let the opposite or dual
labelled poset $P^{\operatorname{opp}}$ have $i<_{P^{\operatorname{opp}}}j$ if
and only if $j<_{P}i$.

For example,

$P=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 32.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 64.5pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{2}$}}}}}}}{\hbox{\kern-5.5pt\raise-36.44443pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{4}$}}}}}}}{\hbox{\kern
32.0pt\raise-36.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
64.5pt\raise-36.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
29.5pt\raise-72.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
67.0pt\raise-72.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\qquad\qquad
P^{\operatorname{opp}}=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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67.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-5.5pt\raise-36.44443pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{4}$}}}}}}}{\hbox{\kern
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3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
64.5pt\raise-36.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
32.0pt\raise-72.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
64.5pt\raise-72.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{2}$}}}}}}}\ignorespaces}}}}\ignorespaces$

The following observation is straightforward.

######  Proposition 5.26.

When $P$ is a linear order corresponding to some permutation
$w=(w_{1},\ldots,w_{n})$ in $\mathfrak{S}_{n}$, then
$w^{\operatorname{opp}}=ww_{0}$ where $w_{0}\in\mathfrak{S}_{n}$ is the
permutation that swaps $i\leftrightarrow n+1-i$ (this is the so-called longest
permutation, thus named due to it having the highest “Coxeter length” among
all permutations in $\mathfrak{S}_{n}$). Furthermore, in this situation one
has $F_{w}({\mathbf{x}})=L_{\alpha}$, that is,
$\operatorname{Des}(w)=D(\alpha)$ if and only if
$\operatorname{Des}(w^{\operatorname{opp}})=D(\omega(\alpha))$, that is
$F_{w^{\operatorname{opp}}}({\mathbf{x}})=L_{\omega(\alpha)}$. Thus,

$S(F_{w}({\mathbf{x}}))=(-1)^{n}F_{w^{\operatorname{opp}}}({\mathbf{x}}).$

For example, given the compositions considered earlier

$\alpha=(4,2,2)=\begin{matrix}&&&&\square&\square\\\ &&&\square&\square&\\\
\square&\square&\square&\square&&\\\
\end{matrix}\qquad\omega(\alpha)=(1,2,2,1,1,1)=\begin{matrix}&&\square\\\
&&\square\\\ &&\square\\\ &\square&\square\\\ \square&\square&\\\ \square&&\\\
\end{matrix}$

if one picks $w=1235\cdot 47\cdot 68$ (with descent positions marked by dots)
having $\operatorname{Des}(w)=\\{4,6\\}=D(\alpha)$, then
$w^{\operatorname{opp}}=ww_{0}=8\cdot 67\cdot 45\cdot 3\cdot 2\cdot 1$ has
$\operatorname{Des}(w^{\operatorname{opp}})=\\{1,3,5,6,7\\}=D(\omega(\alpha))$.

######  Corollary 5.27.

For any labelled poset $P$ on $\\{1,2,\ldots,n\\}$, one has

$S\left(F_{P}({\mathbf{x}})\right)=(-1)^{n}F_{P^{\operatorname{opp}}}({\mathbf{x}}).$

###### Proof.

Since $S$ is linear, one can apply Theorem 5.19 and Proposition 5.26

$S\left(F_{P}({\mathbf{x}})\right)=\sum_{w\in\mathcal{L}(P)}S(F_{w}({\mathbf{x}}))=\sum_{w\in\mathcal{L}(P)}(-1)^{n}F_{w^{\operatorname{opp}}}({\mathbf{x}})=(-1)^{n}F_{P^{\operatorname{opp}}}({\mathbf{x}}).$

as
$\mathcal{L}(P^{\operatorname{opp}})=\\{w^{\operatorname{opp}}:w\in\mathcal{L}(P)\\}$.
∎

######  Remark 5.28.

Malvenuto and Reutenauer, in [119, Theorem 3.1], prove an even more general
antipode formula, which encompasses our Corollary 5.27, Proposition 5.26,
Theorem 5.11 and (5.9). See [67, Theorem 4.2] for a restatement and a self-
contained proof of this theorem (and [67, Theorem 4.6] for an even further
generalization).

We remark on a special case of Corollary 5.27 to which we alluded earlier,
related to skew Schur functions.

######  Corollary 5.29.

In $\Lambda$, the action of $\omega$ and the antipode $S$ on skew Schur
functions $s_{\lambda/\mu}$ are as follows:

(5.11) $\displaystyle\omega(s_{\lambda/\mu})$
$\displaystyle=s_{\lambda^{t}/\mu^{t}}$ (5.12) $\displaystyle
S(s_{\lambda/\mu})$
$\displaystyle=(-1)^{|\lambda/\mu|}s_{\lambda^{t}/\mu^{t}}.$

###### Proof.

Given a skew shape $\lambda/\mu$, one can always create a labelled poset $P$
which is its skew Ferrers poset, together with one of many column-strict
labellings, in such a way that
$F_{P}({\mathbf{x}})=s_{\lambda/\mu}({\mathbf{x}})$. An example is shown here
for $\lambda/\mu=(4,4,2)/(1,1,0)$:

$\lambda/\mu=\begin{matrix}&\square&\square&\square\\\
&\square&\square&\square\\\ \square&\square&&\end{matrix}\qquad\qquad
P=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 13.5pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{5}$}}}}}}}{\hbox{\kern 35.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 54.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 73.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 92.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-5.5pt\raise-20.44443pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{8\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
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13.5pt\raise-40.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{7\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
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The general definition is as follows: Let $P$ be the set of all boxes of the
skew diagram $\lambda/\mu$. Label these boxes by the numbers $1,2,\ldots,n$
(where $n=\left|\lambda/\mu\right|$) row by row from bottom to top (reading
every row from left to right), and then define an order relation $<_{P}$ on
$P$ by requiring that every box be smaller (in $P$) than its right neighbor
and smaller (in $P$) than its lower neighbor. It is not hard to see that in
this situation,
$F_{P^{\operatorname{opp}}}({\mathbf{x}})=\sum_{T}{\mathbf{x}}^{\operatorname{cont}(T)}$
as $T$ ranges over all reverse semistandard tableaux or column-strict plane
partitions of $\lambda^{t}/\mu^{t}$:

$\lambda^{t}/\mu^{t}=\begin{matrix}&&\square\\\ \square&\square&\square&\\\
\square&\square&&\\\ \square&\square&&\end{matrix}\qquad\qquad
P^{\operatorname{opp}}=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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But this means that
$F_{P^{\operatorname{opp}}}({\mathbf{x}})=s_{\lambda^{t}/\mu^{t}}({\mathbf{x}})$,
since the fact that skew Schur functions lie in $\Lambda$ implies that they
can be defined either as generating functions for column-strict tableaux or
reverse semistandard tableaux; see Remark 2.12 above, or [164, Prop. 7.10.4].

Thus we have

$\displaystyle F_{P}({\mathbf{x}})$
$\displaystyle=s_{\lambda/\mu}({\mathbf{x}})$ $\displaystyle
F_{P^{\operatorname{opp}}}({\mathbf{x}})$
$\displaystyle=s_{\lambda^{t}/\mu^{t}}({\mathbf{x}}).$

Proposition 1.46(c) tell us that the antipode for $\operatorname{QSym}$ must
specialize to the antipode for $\Lambda$ (see also Remark 5.43 below), so
(5.12) is a special case of Corollary 5.27. Then (5.11) follows from the
relation (2.30) that $S(f)=(-1)^{n}\omega(f)$ for $f$ in $\Lambda_{n}$. ∎

######  Remark 5.30.

Before leaving $P$-partitions temporarily, we mention two open questions about
them.

The first is a conjecture of Stanley from his thesis [161]. As mentioned in
the proof of Corollary 5.29, each skew Schur function
$s_{\lambda/\mu}({\mathbf{x}})$ is a special instance of $P$-partition
enumerator $F_{P}({\mathbf{x}})$.

######  Conjecture 5.31.

A labelled poset $P$ has $F_{P}({\mathbf{x}})$ symmetric, and not just
quasisymmetric, if and only if $P$ is a column-strict labelling of some skew
Ferrers poset $\lambda/\mu$.

A somewhat weaker result in this direction was proven by Malvenuto in her
thesis [117, Thm. 6.4], showing that if a labelled poset $P$ has the stronger
property that its set of linear extensions $\mathcal{L}(P)$ is a union of
plactic or Knuth equivalence classes, then $P$ must be a column-strict
labelling of a skew Ferrers poset.

The next question is due to P. McNamara, and is suggested by the obvious
factorizations of $P$-partition enumerators $F_{P_{1}\sqcup
P_{2}}({\mathbf{x}})=F_{P_{1}}({\mathbf{x}})F_{P_{2}}({\mathbf{x}})$.

######  Question 5.32.

If $\mathbf{k}$ is a field, does a connected labelled poset $P$ always have
$F_{P}({\mathbf{x}})$ irreducible within the ring $\operatorname{QSym}$?

The phrasing of this question requires further comment. It is assumed here
that ${\mathbf{x}}=(x_{1},x_{2},\ldots)$ is infinite; for example when $P$ is
a $2$-element chain labelled “against the grain” (i.e., the bigger element of
the chain has the smaller label), then
$F_{P}({\mathbf{x}})=e_{2}({\mathbf{x}})$ is irreducible, but its
specialization to two variables ${\mathbf{x}}=(x_{1},x_{2})$ is
$e_{2}(x_{1},x_{2})=x_{1}x_{2}$, which is reducible. If one wishes to work in
finitely many variables ${\mathbf{x}}=(x_{1},\ldots,x_{m})$ one can perhaps
assume that $m$ is at least $|P|+1$.

When working in $\operatorname{QSym}=\operatorname{QSym}({\mathbf{x}})$ in
infinitely many variables, it is perhaps not so clear where factorizations
occur. For example, if $f$ lies in $\operatorname{QSym}$ and factors $f=g\cdot
h$ with $g,h$ in $R({\mathbf{x}})$, does this imply that $g,h$ also lie in
$\operatorname{QSym}$? The answer is “Yes” (for $\mathbf{k}={\mathbb{Z}}$),
but this is not obvious, and was proven by P. Pylyavskyy in [140, Chap. 11].

One also might wonder whether $\operatorname{QSym}_{\mathbb{Z}}$ is a unique
factorization domain, but this follows from the result of M. Hazewinkel ([69]
and [73, Thm. 6.7.5], and Theorem 6.82 further below) who proved a conjecture