F_{w}:=\sum_{k=0}^{n}F_{\operatorname{std}(w_{1},w_{2},\ldots,w_{k})}\otimes
F_{\operatorname{std}(w_{k+1},w_{k+2},\ldots,w_{n})}.$

It is possible to check directly that the maps defined in (8.1) and (8.2)
endow $\operatorname{FQSym}$ with the structure of a graded connected finite
type Hopf algebra; see Hazewinkel, Gubareni, Kirichenko [73, Thm. 7.1.8].
However in justifying this here, we will follow the approach of Duchamp,
Hivert and Thibon [47, §3], which exhibits $\operatorname{FQSym}$ as a
subalgebra of a larger ring of (noncommutative) power series of bounded degree
in a totally ordered alphabet.

######  Definition 8.3.

Given a totally ordered set $I$, create a totally ordered variable set
$\\{X_{i}\\}_{i\in I}$, and the ring $R\langle\\{X_{i}\\}_{i\in I}\rangle$ of
noncommutative power series of bounded degree in this alphabet282282282Let us
recall the definition of $R\langle\\{X_{i}\\}_{i\in I}\rangle$. Let $N$ denote
the free monoid on the alphabet $\\{X_{i}\\}_{i\in I}$; it consists of words
$X_{i_{1}}X_{i_{2}}\cdots X_{i_{k}}$. We define a topological
$\mathbf{k}$-module ${\mathbf{k}}\left<\left<{\\{X_{i}\\}_{i\in
I}}\right>\right>$ to be the Cartesian product $\mathbf{k}^{N}$ (equipped with
the product topology), but we identify its element
$\left(\delta_{w,u}\right)_{u\in N}$ with the word $w$ for every $w\in N$.
Thus, every element $\left(\lambda_{w}\right)_{w\in
N}\in\mathbf{k}^{N}={\mathbf{k}}\left<\left<{\\{X_{i}\\}_{i\in
I}}\right>\right>$ can be rewritten as the convergent sum $\sum_{w\in
N}\lambda_{w}w$. We call $\lambda_{w}$ the coefficient of $w$ in this element
(or the coefficient of this element before $w$). The elements of
${\mathbf{k}}\left<\left<{\\{X_{i}\\}_{i\in I}}\right>\right>$ will be
referred to as noncommutative power series. We define a multiplication on
${\mathbf{k}}\left<\left<{\\{X_{i}\\}_{i\in I}}\right>\right>$ by the formula
$\left(\sum_{w\in N}\lambda_{w}w\right)\left(\sum_{w\in
N}\mu_{w}w\right)=\sum_{w\in N}\left(\sum_{\left(u,v\right)\in N^{2};\
w=uv}\lambda_{u}\mu_{v}\right)w.$ (This is well-defined thanks to the fact
that, for each $w\in N$, there are only finitely many $\left(u,v\right)\in
N^{2}$ satisfying $w=uv$.) Thus, ${\mathbf{k}}\left<\left<{\\{X_{i}\\}_{i\in
I}}\right>\right>$ becomes a $\mathbf{k}$-algebra with unity $1$ (the empty
word). (It is similar to the monoid algebra $\mathbf{k}N$ of $N$ over
$\mathbf{k}$, with the only difference that infinite sums are allowed.) Now,
we define $R\langle\\{X_{i}\\}_{i\in I}\rangle$ to be the
$\mathbf{k}$-subalgebra of ${\mathbf{k}}\left<\left<{\\{X_{i}\\}_{i\in
I}}\right>\right>$ consisting of all noncommutative power series $\sum_{w\in
N}\lambda_{w}w\in{\mathbf{k}}\left<\left<{\\{X_{i}\\}_{i\in I}}\right>\right>$
of bounded degree (i.e., such that all words $w\in N$ of sufficiently high
length satisfy $\lambda_{w}=0$).. Many times, we will use a variable set
${\mathbf{X}}:=(X_{1}<X_{2}<\cdots)$, and call the ring
$R\langle{\mathbf{X}}\rangle$.

We first identify the algebra structure for $\operatorname{FQSym}$ as the
subalgebra of finite type within $R\langle\\{X_{i}\\}_{i\in I}\rangle$ spanned
by the elements

$F_{w}=F_{w}(\\{X_{i}\\}_{i\in
I}):=\sum_{\begin{subarray}{c}{\mathbf{i}}=(i_{1},\ldots,i_{n}):\\\
\operatorname{std}({\mathbf{i}})=w^{-1}\end{subarray}}{\mathbf{X}}_{\mathbf{i}}$

where ${\mathbf{X}}_{{\mathbf{i}}}:=X_{i_{1}}\cdots X_{i_{n}}$, as $w$ ranges
over $\bigcup_{n\geq 0}\mathfrak{S}_{n}$ .

######  Example 8.4.

For the alphabet ${\mathbf{X}}=(X_{1}<X_{2}<\cdots)$, in
$R\langle{\mathbf{X}}\rangle$ one has

$\displaystyle F_{1}$ $\displaystyle=\sum_{1\leq i}X_{i}=X_{1}+X_{2}+\cdots$
$\displaystyle F_{12}$ $\displaystyle=\sum_{1\leq i\leq
j}X_{i}X_{j}=X_{1}^{2}+X_{2}^{2}+\cdots+X_{1}X_{2}+X_{1}X_{3}+X_{2}X_{3}+X_{1}X_{4}+\cdots$
$\displaystyle F_{21}$ $\displaystyle=\sum_{1\leq
i<j}X_{j}X_{i}=X_{2}X_{1}+X_{3}X_{1}+X_{3}X_{2}+X_{4}X_{1}+\cdots$
$\displaystyle F_{312}$
$\displaystyle=\sum_{{\mathbf{i}}:\operatorname{std}({\mathbf{i}})=231}{\mathbf{X}}_{\mathbf{i}}=\sum_{1\leq
i<j\leq k}X_{j}X_{k}X_{i}$
$\displaystyle=X_{2}^{2}X_{1}+X_{3}^{2}X_{1}+X_{3}^{2}X_{2}+\cdots+X_{2}X_{3}X_{1}+X_{2}X_{4}X_{1}+\cdots$

######  Proposition 8.5.

For any totally ordered infinite set $I$, the elements $\\{F_{w}\\}$ as $w$
ranges over $\bigcup_{n\geq 0}\mathfrak{S}_{n}$ form the $\mathbf{k}$-basis
for a subalgebra $\operatorname{FQSym}(\\{X_{i}\\}_{i\in I})$ of
$R\langle{\mathbf{X}}\rangle$, which is graded connected and of finite type,
having multiplication defined $\mathbf{k}$-linearly by (8.1).

Consequently all such algebras are isomorphic to a single algebra
$\operatorname{FQSym}$, having basis $\\{F_{w}\\}$ and multiplication given by
the rule (8.1), with the isomorphism mapping $F_{w}\longmapsto
F_{w}(\\{X_{i}\\}_{i\in I})$.

For example,

$\displaystyle F_{1}F_{21}$
$\displaystyle=(X_{1}+X_{2}+X_{3}+\cdots)(X_{2}X_{1}+X_{3}X_{1}+X_{3}X_{2}+X_{4}X_{1}+\cdots)$
$\displaystyle=X_{1}\cdot X_{3}X_{2}+X_{1}\cdot X_{4}X_{2}+\cdots+X_{1}\cdot
X_{2}X_{1}+X_{2}\cdot X_{3}X_{2}+X_{2}\cdot X_{4}X_{2}+\cdots$
$\displaystyle\qquad+X_{2}\cdot X_{3}X_{1}+X_{2}\cdot
X_{4}X_{1}+\cdots+X_{2}\cdot X_{2}X_{1}+X_{3}\cdot X_{3}X_{1}+X_{3}\cdot
X_{3}X_{2}+\cdots$ $\displaystyle\qquad+X_{3}\cdot X_{2}X_{1}+X_{4}\cdot
X_{2}X_{1}+\cdots$
$\displaystyle=\sum_{{\mathbf{i}}:\operatorname{std}({\mathbf{i}})=132}{\mathbf{X}}_{\mathbf{i}}+\sum_{{\mathbf{i}}:\operatorname{std}({\mathbf{i}})=231}{\mathbf{X}}_{\mathbf{i}}+\sum_{{\mathbf{i}}:\operatorname{std}({\mathbf{i}})=321}{\mathbf{X}}_{\mathbf{i}}=F_{132}+F_{312}+F_{321}=\sum_{w\in{1}\,\shuffle\,{32}}F_{w}$

###### Proof of Proposition 8.5.

The elements $\\{F_{w}(\\{X_{i}\\}_{i\in I})\\}$ are linearly independent as
they are supported on disjoint monomials, and so form a $\mathbf{k}$-basis for
their span. The fact that they multiply via rule (8.1) is the equivalence of
conditions (i) and (iii) in the following Lemma 8.6, from which all the
remaining assertions follow. ∎

######  Lemma 8.6.

For a triple of permutations

$\displaystyle u$ $\displaystyle=(u_{1},\ldots,u_{k})\text{ in
}\mathfrak{S}_{k},$ $\displaystyle v$
$\displaystyle=(v_{1},\ldots,v_{n-k})\text{ in }\mathfrak{S}_{n-k},$
$\displaystyle w$ $\displaystyle=(w_{1},\ldots,w_{n})\text{ in
}\mathfrak{S}_{n},$

the following conditions are equivalent:

  1. (i)

$w^{-1}$ lies in the set ${u^{-1}}\,\shuffle\,{v^{-1}[k]}$.

  2. (ii)

$u=\operatorname{std}(w_{1},\ldots,w_{k})$ and
$v=\operatorname{std}(w_{k+1},\ldots,w_{n})$,

  3. (iii)

for some word ${\mathbf{i}}=(i_{1},\ldots,i_{n})$ with
$\operatorname{std}({\mathbf{i}})=w$ one has
$u=\operatorname{std}(i_{1},\ldots,i_{k})$ and
$v=\operatorname{std}(i_{k+1},\ldots,i_{n})$.

###### Proof of Lemma 8.6..

The implication (ii) $\Rightarrow$ (iii) is clear since
$\operatorname{std}(w)=w$. The reverse implication (iii) $\Rightarrow$ (ii) is
best illustrated by example, e.g. considering Example 8.2 as concatenated,
with $n=11,k=6,n-k=5$:

w=std | (b | a | c | c | b | a | — |  | a | b | a | c | b)
---|---|---|---|---|---|---|---|---|---|---|---|---|---
= | (5 | 1 | 9 | 10 | 6 | 2 | — |  | 3 | 7 | 4 | 11 | 8)
u=std | (5 | 1 | 9 | 10 | 6 | 2) |  | v = std | (3 | 7 | 4 | 11 | 8)
= | (3 | 1 | 5 | 6 | 4 | 2) |  | = | (1 | 3 | 2 | 5 | 4)
=std | (b | a | c | c | b | a) |  | = std | (a | b | a | c | b)

The equivalence of (i) and (ii) is a fairly standard consequence of unique
parabolic factorization $W=W^{J}W_{J}$ where $W=\mathfrak{S}_{n}$ and
$W_{J}=\mathfrak{S}_{k}\times\mathfrak{S}_{n-k}$, so that $W^{J}$ are the
minimum-length coset representatives for cosets $xW_{J}$ (that is, the
permutations $x\in\mathfrak{S}_{n}$ satisfying $x_{1}<\cdots<x_{k}$ and
$x_{k+1}<\cdots<x_{n}$). One can uniquely express any $w$ in $W$ as $w=xy$
with $x$ in $W^{J}$ and $y$ in $W_{J}$, which here means that $y=u\cdot
v[k]=v[k]\cdot u$ for some $u$ in $\mathfrak{S}_{k}$ and $v$ in
$\mathfrak{S}_{n-k}$. Therefore $w=xuv[k]$, if and only if
$w^{-1}=u^{-1}v^{-1}[k]x^{-1}$, which means that $w^{-1}$ is the shuffle of
the sequences $u^{-1}$ in positions $\\{x_{1},\ldots,x_{k}\\}$ and $v^{-1}[k]$
in positions $\\{x_{k+1},\ldots,x_{n}\\}$. ∎

######  Example 8.7.

To illustrate the equivalence of (i) and (ii) and the parabolic factorization
in the preceding proof, let $n=9$ and $k=5$ with

$\displaystyle w$ $\displaystyle=\left(\begin{matrix}1&2&3&4&5&|&6&7&8&9\\\
4&9&6&1&5&|&8&2&3&7\end{matrix}\right)$
$\displaystyle=\left(\begin{matrix}1&2&3&4&5&|&6&7&8&9\\\
1&4&5&6&9&|&2&3&7&8\end{matrix}\right)\left(\begin{matrix}1&2&3&4&5\\\
2&5&4&1&3\end{matrix}\right)\left(\begin{matrix}6&7&8&9\\\
9&6&7&8\end{matrix}\right)$ $\displaystyle=x\cdot u\cdot v[k]$ $\displaystyle
w^{-1}$ $\displaystyle=\left(\begin{matrix}1&2&3&4&5&6&7&8&9\\\
4&9&6&1&5&8&2&3&7\end{matrix}\right)$
$\displaystyle=\left(\begin{matrix}1&2&3&4&5\\\
4&1&5&3&2\end{matrix}\right)\left(\begin{matrix}6&7&8&9\\\
7&8&9&6\end{matrix}\right)\left(\begin{matrix}1&2&3&4&5&6&7&8&9\\\
\underline{1}&\underline{\underline{6}}&\underline{\underline{7}}&\underline{2}&\underline{3}&\underline{4}&\underline{\underline{8}}&\underline{\underline{9}}&\underline{5}\end{matrix}\right)$
$\displaystyle=u^{-1}\cdot v^{-1}[k]\cdot x^{-1}$

Proposition 8.5 yields that $\operatorname{FQSym}$ is isomorphic to the
$\mathbf{k}$-subalgebra $\operatorname{FQSym}\left({\mathbf{X}}\right)$ of the
$\mathbf{k}$-algebra $R\left<{\mathbf{X}}\right>$ when ${\mathbf{X}}$ is the
variable set $\left(X_{1}<X_{2}<\cdots\right)$. We identify
$\operatorname{FQSym}$ with $\operatorname{FQSym}\left({\mathbf{X}}\right)$
along this isomorphism. For any infinite alphabet
$\left\\{X_{i}\right\\}_{i\in I}$ and any $f\in\operatorname{FQSym}$, we
denote by $f\left(\left\\{x_{i}\right\\}_{i\in I}\right)$ the image of $f$
under the algebra isomorphism
$\operatorname{FQSym}\to\operatorname{FQSym}\left(\left\\{X_{i}\right\\}_{i\in
I}\right)$ defined in Proposition 8.5.

One can now use this to define a coalgebra structure on
$\operatorname{FQSym}$. Roughly speaking, one wants to first evaluate an
element $f$ in
$\operatorname{FQSym}\cong\operatorname{FQSym}\left({\mathbf{X}}\right)\cong\operatorname{FQSym}\left({\mathbf{X}},{\mathbf{Y}}\right)$
as $f({\mathbf{X}},{\mathbf{Y}})$, using the linearly ordered variable set
$({\mathbf{X}},{\mathbf{Y}}):=(X_{1}<X_{2}<\cdots<Y_{1}<Y_{2}<\cdots)$. Then
one should take the image of $f({\mathbf{X}},{\mathbf{Y}})$ after imposing the
partial commutativity relations

(8.3) $X_{i}Y_{j}=Y_{j}X_{i}\text{ for every pair
}(X_{i},Y_{j})\in{\mathbf{X}}\times{\mathbf{Y}},$

and hope that this image lies in a subalgebra isomorphic to

$\operatorname{FQSym}\left({\mathbf{X}}\right)\otimes\operatorname{FQSym}\left({\mathbf{Y}}\right)\cong\operatorname{FQSym}\otimes\operatorname{FQSym}.$

We argue this somewhat carefully. Start by considering the canonical monoid
epimorphism

(8.4)
$F\langle{\mathbf{X}},{\mathbf{Y}}\rangle\overset{\rho}{\twoheadrightarrow}M,$

where $F\langle{\mathbf{X}},{\mathbf{Y}}\rangle$ denotes the free monoid on
the alphabet $({\mathbf{X}},{\mathbf{Y}})$ and $M$ denotes its quotient monoid
imposing the partial commutativity relations (8.3). Let $\mathbf{k}^{M}$
denote the $\mathbf{k}$-module of all functions $f:M\rightarrow\mathbf{k}$,
with pointwise addition and scalar multiplication; similarly define
$\mathbf{k}^{F\langle{\mathbf{X}},{\mathbf{Y}}\rangle}$. As both monoids
$F\langle{\mathbf{X}},{\mathbf{Y}}\rangle$ and $M$ enjoy the property that an
element $m$ has only finitely many factorizations as $m=m_{1}m_{2}$, one can
define a convolution algebra structure on both
$\mathbf{k}^{F\langle{\mathbf{X}},{\mathbf{Y}}\rangle}$ and $\mathbf{k}^{M}$
via

$(f_{1}\star f_{2})(m)=\sum_{\begin{subarray}{c}(m_{1},m_{2})\in N\times N:\\\
m=m_{1}m_{2}\end{subarray}}f_{1}(m_{1})f_{2}(m_{2}),$

where $N$ is respectively $F\langle{\mathbf{X}},{\mathbf{Y}}\rangle$ or $M$.
As fibers of the map $\rho$ in (8.4) are finite, it induces a map of
convolution algebras, which we also call $\rho$:

(8.5)
$\mathbf{k}^{F\langle{\mathbf{X}},{\mathbf{Y}}\rangle}\overset{\rho}{\twoheadrightarrow}\mathbf{k}^{M}.$

Now recall that $R\langle{\mathbf{X}}\rangle$ denotes the algebra of
noncommutative formal power series in the variable set ${\mathbf{X}}$, of
bounded degree, with coefficients in $\mathbf{k}$. One similarly has the ring
$R\langle{\mathbf{X}},{\mathbf{Y}}\rangle$, which can be identified with the
subalgebra of $\mathbf{k}^{F\langle{\mathbf{X}},{\mathbf{Y}}\rangle}$
consisting of the functions
$f:F\langle{\mathbf{X}},{\mathbf{Y}}\rangle\rightarrow\mathbf{k}$ having a
bound on the length of the words in their support (the value of $f$ on a word
in $({\mathbf{X}},{\mathbf{Y}})$ gives its power series coefficient
corresponding to said word). We let $R\langle M\rangle$ denote the analogous
subalgebra of $\mathbf{k}^{M}$; this can be thought of as the algebra of
bounded degree “partially commutative power series” in the variable sets
${\mathbf{X}}$ and ${\mathbf{Y}}$. Note that $\rho$ restricts to a map

(8.6)
$R\langle{\mathbf{X}},{\mathbf{Y}}\rangle\overset{\rho}{\rightarrow}R\langle
M\rangle.$

Finally, we claim (and see Proposition 8.9 below for a proof) that this
further restricts to a map

(8.7)
$\operatorname{FQSym}\left({\mathbf{X}},{\mathbf{Y}}\right)\overset{\rho}{\rightarrow}\operatorname{FQSym}\left({\mathbf{X}}\right)\otimes\operatorname{FQSym}\left({\mathbf{Y}}\right)$

in which the target is identified with its image under the
(injective283283283as images of the basis $F_{u}({\mathbf{X}})\otimes
F_{v}({\mathbf{Y}})$ of
$\operatorname{FQSym}({\mathbf{X}})\otimes\operatorname{FQSym}({\mathbf{Y}})$
are supported on disjoint monomials in $R\langle M\rangle$, so linearly
independent.) multiplication map

$\begin{array}[]{rcl}\operatorname{FQSym}\left({\mathbf{X}}\right)\otimes\operatorname{FQSym}\left({\mathbf{Y}}\right)&\hookrightarrow&R\langle
M\rangle\\\ f({\mathbf{X}})\otimes
g({\mathbf{Y}})&\mapsto&f({\mathbf{X}})g({\mathbf{Y}}).\end{array}$

Using the identification of $\operatorname{FQSym}$ with all three of
$\operatorname{FQSym}\left({\mathbf{X}}\right),\operatorname{FQSym}\left({\mathbf{Y}}\right),\operatorname{FQSym}\left({\mathbf{X}},{\mathbf{Y}}\right)$,
the map $\rho$ in (8.7) will then define a coproduct structure on
$\operatorname{FQSym}$. Abusing notation, for $f$ in $\operatorname{FQSym}$,
we will simply write $\Delta(f)=f({\mathbf{X}},{\mathbf{Y}})$ instead of
$\rho(f({\mathbf{X}},{\mathbf{Y}}))$.

######  Example 8.8.

Recall from Example 8.4 that one has

$F_{312}=\sum_{{\mathbf{i}}:\operatorname{std}({\mathbf{i}})=231}{\mathbf{X}}_{\mathbf{i}}=\sum_{1\leq
i<j\leq k}X_{j}X_{k}X_{i}$

and therefore its coproduct is

$\displaystyle\Delta F_{312}$
$\displaystyle=F_{312}(X_{1},X_{2},\ldots,Y_{1},Y_{2},\ldots)\qquad\qquad\left(\text{by
our abuse of notation}\right)$ $\displaystyle=\sum_{i<j\leq
k}X_{j}X_{k}X_{i}+\sum_{\begin{subarray}{c}i<j,\\\
k\end{subarray}}X_{j}Y_{k}X_{i}+\sum_{\begin{subarray}{c}i,\\\ j\leq
k\end{subarray}}Y_{j}Y_{k}X_{i}+\sum_{i<j\leq k}Y_{j}Y_{k}Y_{i}$
$\displaystyle=\sum_{i<j\leq k}X_{j}X_{k}X_{i}\cdot
1+\sum_{\begin{subarray}{c}i<j,\\\ k\end{subarray}}X_{j}X_{i}\cdot
Y_{k}+\sum_{\begin{subarray}{c}i,\\\ j\leq k\end{subarray}}X_{i}\cdot
Y_{j}Y_{k}+\sum_{i<j\leq k}1\cdot Y_{j}Y_{k}Y_{i}$
$\displaystyle=F_{312}({\mathbf{X}})\cdot 1+F_{21}({\mathbf{X}})\cdot
F_{1}({\mathbf{Y}})+F_{1}({\mathbf{X}})\cdot F_{12}({\mathbf{Y}})+1\cdot
F_{312}({\mathbf{Y}})$ $\displaystyle=F_{312}\otimes 1+F_{21}\otimes
F_{1}+F_{1}\otimes F_{12}+1\otimes F_{312}.$

######  Proposition 8.9.

The map $\rho$ in (8.6) does restrict as claimed to a map as in (8.7), and
hence defines a coproduct on $\operatorname{FQSym}$, acting on the
$\\{F_{w}\\}$ basis by the rule (8.2). This endows $\operatorname{FQSym}$ with
the structure of a graded connected finite type Hopf algebra.

###### Proof.

Let $I$ be the totally ordered set $\left\\{1<2<3<\cdots\right\\}$. Let $J$ be
the totally ordered set
$\left\\{1<2<3<\cdots<\widetilde{1}<\widetilde{2}<\widetilde{3}<\cdots\right\\}$.
We set $X_{\widetilde{i}}=Y_{i}$ for every positive integer $i$. Then, the
alphabet $({\mathbf{X}},{\mathbf{Y}})$ can be written as
$\left\\{X_{i}\right\\}_{i\in J}$.

If ${\mathbf{i}}$ is a word over the alphabet
$I=\left\\{1<2<3<\cdots\right\\}$, then we denote by
$\widetilde{{\mathbf{i}}}$ the word over $J$ obtained from ${\mathbf{i}}$ by
replacing every letter $i$ by $\widetilde{i}$.

For the first assertion of Proposition 8.9, it suffices to check that $F_{w}$
indeed has the image under $\Delta$ claimed in (8.2). Let $n\in{\mathbb{N}}$
and $w\in\mathfrak{S}_{n}$. Then,

$\displaystyle\Delta F_{w}$
$\displaystyle=F_{w}\left({\mathbf{X}},{\mathbf{Y}}\right)\qquad\qquad\left(\text{by
our abuse of notation}\right)$ $\displaystyle=\sum_{{\mathbf{i}}\in
J^{n}:\operatorname{std}({\mathbf{i}})=w^{-1}}({\mathbf{X}},{\mathbf{Y}})_{{\mathbf{i}}}=\sum_{\mathbf{t}\in
J^{n}:\operatorname{std}(\mathbf{t})=w^{-1}}({\mathbf{X}},{\mathbf{Y}})_{\mathbf{t}}$
(8.8) $\displaystyle=\sum_{k=0}^{n}\sum_{({\mathbf{i}},{\mathbf{j}})\in
I^{k}\times I^{n-k}}\sum_{\begin{subarray}{c}\mathbf{t}\in J^{n}:\\\
\operatorname{std}(\mathbf{t})=w^{-1};\\\
\mathbf{t}\in{{\mathbf{i}}}\,\shuffle\,{\widetilde{{\mathbf{j}}}}\end{subarray}}({\mathbf{X}},{\mathbf{Y}})_{\mathbf{t}}$

(since for every $\mathbf{t}\in J^{n}$, there exists exactly one choice of
$k\in\left\\{0,1,\ldots,n\right\\}$ and
$\left({\mathbf{i}},{\mathbf{j}}\right)\in I^{k}\times I^{n-k}$ satisfying
$\mathbf{t}\in{{\mathbf{i}}}\,\shuffle\,{\widetilde{{\mathbf{j}}}}$; namely,
${\mathbf{i}}$ is the restriction of $\mathbf{t}$ to the subalphabet $I$ of
$J$, whereas ${\mathbf{j}}$ is the restriction of $\mathbf{t}$ to $J\setminus
I$, and $k$ is the length of ${\mathbf{i}}$).

We now fix $k$ and $\left({\mathbf{i}},{\mathbf{j}}\right)$, and try to
simplify the inner sum $\sum_{\begin{subarray}{c}\mathbf{t}\in J^{n}:\\\
\operatorname{std}(\mathbf{t})=w^{-1};\\\
\mathbf{t}\in{{\mathbf{i}}}\,\shuffle\,{\widetilde{{\mathbf{j}}}}\end{subarray}}({\mathbf{X}},{\mathbf{Y}})_{\mathbf{t}}$
on the right hand side of (8.8). First we notice that this sum is nonempty if
and only if there exists some
$\mathbf{t}\in{{\mathbf{i}}}\,\shuffle\,{\widetilde{{\mathbf{j}}}}$ satisfying
$\operatorname{std}(\mathbf{t})=w^{-1}$. This existence is easily seen to be
equivalent to
$w^{-1}\in{\operatorname{std}({\mathbf{i}})}\,\shuffle\,{\operatorname{std}({\mathbf{j}})[k]}$
(since the standardization of any shuffle in
${{\mathbf{i}}}\,\shuffle\,{\widetilde{{\mathbf{j}}}}$ is the corresponding
shuffle in
${\operatorname{std}({\mathbf{i}})}\,\shuffle\,{\operatorname{std}({\mathbf{j}})[k]}$).
This, in turn, is equivalent to
$\operatorname{std}({\mathbf{i}})=(\operatorname{std}(w_{1},\ldots,w_{k}))^{-1}$
and
$\operatorname{std}({\mathbf{j}})=(\operatorname{std}(w_{k+1},\ldots,w_{n}))^{-1}$
(according to the equivalence (i) $\Longleftrightarrow$ (ii) in Lemma 8.6).
Hence, the inner sum on the right hand side of (8.8) is nonempty if and only
if
$\operatorname{std}({\mathbf{i}})=(\operatorname{std}(w_{1},\ldots,w_{k}))^{-1}$
and
$\operatorname{std}({\mathbf{j}})=(\operatorname{std}(w_{k+1},\ldots,w_{n}))^{-1}$.
When it is nonempty, it has only one addend284284284In fact, the elements
$\operatorname{std}\left(\mathbf{t}\right)$ for
$\mathbf{t}\in{{\mathbf{i}}}\,\shuffle\,{\widetilde{{\mathbf{j}}}}$ are
pairwise distinct, and thus only one of them can equal $w^{-1}$., and this
addend is
$\left({\mathbf{X}},{\mathbf{Y}}\right)_{\mathbf{t}}={\mathbf{X}}_{\mathbf{i}}{\mathbf{Y}}_{\mathbf{j}}$
(since $\mathbf{t}\in{{\mathbf{i}}}\,\shuffle\,{\widetilde{{\mathbf{j}}}}$).
Summarizing, we see that the inner sum on the right hand side of (8.8) equals
${\mathbf{X}}_{\mathbf{i}}{\mathbf{Y}}_{\mathbf{j}}$ when
$\operatorname{std}({\mathbf{i}})=(\operatorname{std}(w_{1},\ldots,w_{k}))^{-1}$
and
$\operatorname{std}({\mathbf{j}})=(\operatorname{std}(w_{k+1},\ldots,w_{n}))^{-1}$,
and is empty otherwise. Thus, (8.8) simplifies to

$\displaystyle\Delta F_{w}$
$\displaystyle=\sum_{k=0}^{n}\sum_{\begin{subarray}{c}({\mathbf{i}},{\mathbf{j}})\in
I^{k}\times I^{n-k}:\\\
\operatorname{std}({\mathbf{i}})=(\operatorname{std}(w_{1},\ldots,w_{k}))^{-1}\\\
\operatorname{std}({\mathbf{j}})=(\operatorname{std}(w_{k+1},\ldots,w_{n}))^{-1}\end{subarray}}{\mathbf{X}}_{\mathbf{i}}{\mathbf{Y}}_{\mathbf{j}}$
$\displaystyle=\sum_{k=0}^{n}F_{\operatorname{std}(w_{1},\ldots,w_{k})}({\mathbf{X}})F_{\operatorname{std}(w_{k+1},\ldots,w_{n})}({\mathbf{Y}})$
$\displaystyle=\sum_{k=0}^{n}F_{\operatorname{std}(w_{1},\ldots,w_{k})}\otimes
F_{\operatorname{std}(w_{k+1},\ldots,w_{n})}\in\operatorname{FQSym}\otimes\operatorname{FQSym}.$

This proves (8.2), and thus the first assertion of Proposition 8.9.

From this, it is easy to derive that $\Delta$ satisfies coassociativity (i.e.,
the diagram (1.3) holds for $C=\operatorname{FQSym}$). (Alternatively, one can
obtain this from the associativity of multiplication using Corollary 8.11.) We
have already verified the rule (8.2). The connected graded structure on
$\operatorname{FQSym}$ gives a counit and an antipode for free. ∎

######  Exercise 8.10.

We say that a permutation $w\in\mathfrak{S}_{n}$ is connected if $n$ is a
positive integer and if there exists no $i\in\left\\{1,2,\ldots,n-1\right\\}$
satisfying
$f\left(\left\\{1,2,\ldots,i\right\\}\right)=\left\\{1,2,\ldots,i\right\\}$.
Let $\mathfrak{CS}$ denote the set of all connected permutations of all
$n\in{\mathbb{N}}$. Show that $\operatorname{FQSym}$ is a free
(noncommutative) $\mathbf{k}$-algebra with generators
$\left(F_{w}\right)_{w\in\mathfrak{CS}}$. (This statement means that
$\left(F_{w_{1}}F_{w_{2}}\cdots F_{w_{k}}\right)_{k\in{\mathbb{N}};\
\left(w_{1},w_{2},\ldots,w_{k}\right)\in\mathfrak{CS}^{k}}$ is a basis of the
$\mathbf{k}$-module $\operatorname{FQSym}$.)

[Hint: This is a result of Poirier and Reutenauer [138, Theorem 2.1]; it is
much easier than the similar Theorem 6.82.]

######  Corollary 8.11.

The Hopf algebra $\operatorname{FQSym}$ is self-dual: Let $\\{G_{w}\\}$ be the
dual $\mathbf{k}$-basis to the $\mathbf{k}$-basis $\\{F_{w}\\}$ for
$\operatorname{FQSym}$. Then, the $\mathbf{k}$-linear map sending
$G_{w}\longmapsto F_{w^{-1}}$ is a Hopf algebra isomorphism
$\operatorname{FQSym}^{o}\longrightarrow\operatorname{FQSym}$.

###### Proof.

For any $0\leq k\leq n$, any $u\in\mathfrak{S}_{k}$ and any
$v\in\mathfrak{S}_{n-k}$, one has

$F_{u^{-1}}F_{v^{-1}}=\sum_{w^{-1}\in{u^{-1}}\,\shuffle\,{v^{-1}[k]}}F_{w^{-1}}=\sum_{\begin{subarray}{c}w:\\\
\operatorname{std}(w_{1},\ldots,w_{k})=u\\\
\operatorname{std}(w_{k+1},\ldots,w_{n})=v\end{subarray}}F_{w^{-1}}$

via the equivalence of (i) and (ii) in Lemma 8.6. On the other hand, in
$\operatorname{FQSym}^{o}$, the dual $\mathbf{k}$-basis $\\{G_{w}\\}$ to the
$\mathbf{k}$-basis $\\{F_{w}\\}$ for $\operatorname{FQSym}$ should have
product formula

$G_{u}G_{v}=\sum_{\begin{subarray}{c}w:\\\
\operatorname{std}(w_{1},\ldots,w_{k})=u\\\
\operatorname{std}(w_{k+1},\ldots,w_{n})=v\end{subarray}}G_{w}$

coming from the coproduct formula (8.2) for $\operatorname{FQSym}$ in the
$\\{F_{w}\\}$-basis. Comparing these equalities, we see that the
$\mathbf{k}$-linear map $\tau$ sending $G_{w}\longmapsto F_{w^{-1}}$ is an
isomorphism $\operatorname{FQSym}^{o}\longrightarrow\operatorname{FQSym}$ of
$\mathbf{k}$-algebras. Hence, the adjoint
$\tau^{\ast}:\operatorname{FQSym}^{o}\to\left(\operatorname{FQSym}^{o}\right)^{o}$
of this map is an isomorphism of $\mathbf{k}$-coalgebras. But identifying
$\left(\operatorname{FQSym}^{o}\right)^{o}$ with $\operatorname{FQSym}$ in the
natural way (since $\operatorname{FQSym}$ is of finite type), we easily see
that $\tau^{\ast}=\tau$, whence $\tau$ itself is an isomorphism of both
$\mathbf{k}$-algebras and $\mathbf{k}$-coalgebras, hence of
$\mathbf{k}$-bialgebras, hence of Hopf algebras. ∎

We can now be a bit more precise about the relations between the various
algebras

$\Lambda,\operatorname{QSym},\operatorname{NSym},\operatorname{FQSym},R\langle{\mathbf{X}}\rangle,R({\mathbf{x}}).$

Not only does $\operatorname{FQSym}$ allow one to lift the Hopf structure of
$\operatorname{QSym}$, it dually allows one to extend the Hopf structure of
$\operatorname{NSym}$. To set up this duality, note that Corollary 8.11
motivates the choice of an inner product on $\operatorname{FQSym}$ in which

$(F_{u},F_{v}):=\delta_{u^{-1},v}.$

We wish to identify the images of the ribbon basis $\\{R_{\alpha}\\}$ of
$\operatorname{NSym}$ when included in $\operatorname{FQSym}$.

######  Definition 8.12.

For any composition $\alpha$, define an element ${\mathbf{R}}_{\alpha}$ of
$\operatorname{FQSym}$ by

${\mathbf{R}}_{\alpha}:=\sum_{w:\operatorname{Des}(w)=D(\alpha)}F_{w^{-1}}=\sum_{\begin{subarray}{c}(w,{\mathbf{i}}):\\\
\operatorname{Des}(w)=D(\alpha)\\\
\operatorname{std}({\mathbf{i}})=w\end{subarray}}{\mathbf{X}}_{\mathbf{i}}=\sum_{{\mathbf{i}}:\operatorname{Des}({\mathbf{i}})=D(\alpha)}{\mathbf{X}}_{\mathbf{i}}$

where the $w$ in the sums are supposed to belong to
$\mathfrak{S}_{\left|\alpha\right|}$, and where the descent set of a sequence
${\mathbf{i}}=(i_{1},\ldots,i_{n})$ is defined by

$\operatorname{Des}({\mathbf{i}}):=\\{j\in\\{1,2,\ldots,n-1\\}:i_{j}>i_{j+1}\\}=\operatorname{Des}(\operatorname{std}({\mathbf{i}})).$

Alternatively, ${\mathbf{R}}_{\alpha}=\sum_{T}{\mathbf{X}}_{T}$ in which the
sum is over column-strict tableaux of the ribbon skew shape $\alpha$, and
${\mathbf{X}}_{T}={\mathbf{X}}_{{\mathbf{i}}}$ in which ${\mathbf{i}}$ is the
sequence of entries of $T$ read in order from the southwest toward the
northeast.

######  Example 8.13.

Taking $\alpha=(1,3,2)$, with ribbon shape and column-strict fillings $T$ as
shown

$\begin{matrix}&&\square&\square\\\ \square&\square&\square&\\\
\square&&&\end{matrix}\qquad T=\quad\begin{matrix}&&&&i_{5}&\leq&i_{6}\\\
&&&&\wedge&&\\\ i_{2}&\leq&i_{3}&\leq&i_{4}&\\\ \wedge&&&&&\\\
i_{1}&&&&&\end{matrix}$

one has that

${\mathbf{R}}_{(1,3,2)}=\sum_{\begin{subarray}{c}{\mathbf{i}}=(i_{1},i_{2},i_{3},i_{4},i_{5},i_{6}):\\\
\operatorname{Des}({\mathbf{i}})=D(\alpha)=\\{1,4\\}\end{subarray}}{\mathbf{X}}_{\mathbf{i}}=\sum_{i_{1}>i_{2}\leq
i_{3}\leq i_{4}>i_{5}\leq
i_{6}}X_{i_{1}}X_{i_{2}}X_{i_{3}}X_{i_{4}}X_{i_{5}}X_{i_{6}}=\sum_{T}{\mathbf{X}}_{T}$

######  Corollary 8.14.

For every $n\in{\mathbb{N}}$ and $w\in\mathfrak{S}_{n}$, we let $\gamma(w)$
denote the unique composition $\alpha$ of $n$ satisfying
$D\left(\alpha\right)=\operatorname{Des}\left(w\right)$. The following
${\mathbb{Z}}$-linear surjection and injection

$\begin{array}[]{rcl}\operatorname{FQSym}&\overset{\pi}{\twoheadrightarrow}&\operatorname{QSym}\\\
F_{w}&\longmapsto&L_{\gamma(w)}\\\ &&\\\
\operatorname{NSym}&\overset{\iota}{\hookrightarrow}&\operatorname{FQSym}\\\
R_{\alpha}&\longmapsto&{\mathbf{R}}_{\alpha}\end{array}$

are Hopf morphisms, and are also adjoint maps with respect to the above choice
of inner product on $\operatorname{FQSym}$ and the usual dual pairing between
$\operatorname{NSym}$ and $\operatorname{QSym}$. Furthermore, the second map
$\iota$ lets one factor the surjection
$\operatorname{NSym}\twoheadrightarrow\Lambda$ as follows

$\begin{array}[]{rcccl}\operatorname{NSym}&\rightarrow&\operatorname{FQSym}\hookrightarrow
R\langle{\mathbf{X}}\rangle&\rightarrow&R({\mathbf{x}})\\\
R_{\alpha}&\longmapsto&{\mathbf{R}}_{\alpha}&\longmapsto&s_{\alpha}({\mathbf{x}})\end{array}$

through the abelianization $R\langle{\mathbf{X}}\rangle\twoheadrightarrow
R({\mathbf{x}})$ sending the noncommutative variable $X_{i}$ to the
commutative $x_{i}$.

###### Proof.

The fact that
$\operatorname{FQSym}\overset{\pi}{\twoheadrightarrow}\operatorname{QSym}$ is
a Hopf map comes from checking that it respects the product (compare (5.8) and
(8.1)) and the coproducts (compare (5.7) and (8.2)) then applying Proposition
1.46(c). It follows that
$\operatorname{NSym}\overset{\iota}{\hookrightarrow}\operatorname{FQSym}$ is a
Hopf map once we check the adjointness assertion, via the following
calculation:

$\displaystyle(\iota(R_{\alpha}),F_{w})=({\mathbf{R}}_{\alpha},F_{w})=\sum_{u:\operatorname{Des}(u)=D(\alpha)}(F_{u^{-1}},F_{w})$
$\displaystyle=\left\\{\begin{array}[]{rl}1&\text{ if
}\operatorname{Des}(w)=D(\alpha)\\\ 0&\text{ otherwise }\end{array}\right\\}$
$\displaystyle=(R_{\alpha},L_{\gamma(w)})=(R_{\alpha},\pi(F_{w})).$

The last assertion is clear: the abelianization map sends the noncommutative
tableau monomial ${\mathbf{X}}_{T}$ to the commutative tableau monomial
${\mathbf{x}}_{T}$. ∎

We summarize some of this picture as follows:

$\textstyle{\operatorname{FQSym}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$dual$\textstyle{\operatorname{FQSym}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\operatorname{NSym}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$dual$\scriptstyle{\pi}$$\scriptstyle{\iota}$$\textstyle{\operatorname{QSym}}$$\textstyle{\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$dual$\textstyle{\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Furthermore, if we denote by $\iota$ the canonical inclusion
$\Lambda\to\operatorname{QSym}$ as well, then the diagram

$\textstyle{\operatorname{FQSym}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\operatorname{NSym}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\iota}$$\textstyle{\operatorname{QSym}}$$\textstyle{\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$

is commutative (according to the last sentence of Corollary 8.14).

######  Remark 8.15.

Different notations for $\operatorname{FQSym}$ appear in the literature. In
the book [22] (which presents an unusual approach to the character theory of
the symmetric group using $\operatorname{FQSym}$), the Hopf algebra
$\operatorname{FQSym}$ is called $\mathcal{P}$, and its basis that we call
$\left\\{G_{w}\right\\}_{w\in\mathfrak{S}_{n}}$ is denoted
$\left\\{w\right\\}_{w\in\mathfrak{S}_{n}}$. In [73, Chapter 7], the Hopf
algebra $\operatorname{FQSym}$ and its basis
$\left\\{F_{w}\right\\}_{w\in\mathfrak{S}_{n}}$ are denoted $MPR$ and
$\left\\{w\right\\}_{w\in\mathfrak{S}_{n}}$, respectively.

##  9\. Further topics

The following is a list of topics that were, at one point, planned to be
touched in class, but did not make the cut. They might get elaborated upon in
a future version of these notes.

####  9.0.1. $0$-Hecke algebras

  * •

Review of representation theory of finite-dimensional algebras.

Review the notions of indecomposables, simples, projectives, along with the
theorems of Krull-Remak-Schmidt, of Jordan-Hölder, and the two kinds of
Grothendieck groups dual to each other.

  * •

$0$-Hecke algebra representation theory.

Describe the simples and projectives, following Denton, Hivert, Schilling,
Thiery [43] on ${\mathcal{J}}$-trivial monoids.

  * •

Nsym and Qsym as Grothendieck groups.

Give Krob and Thibon’s interpretation (see [173, §5] for a brief summary) of

    1. $\bullet$

$\operatorname{QSym}$ and the Grothendieck group of composition series, and

    2. $\bullet$

$\operatorname{NSym}$ and the Grothendieck group of projectives.

######  Remark 9.1.

Mention P. McNamara’s interpretation, in the case of supersolvable lattices,
of the Ehrenborg quasisymmetric function as the composition series enumerator
for an $H_{n}(0)$-action on the maximal chains

####  9.0.2. Aguiar-Bergeron-Sottile character theory Part II: Odd and even
characters, subalgebras

####  9.0.3. Face enumeration, Eulerian posets, and cd-indices

Borrowing from Billera’s ICM notes.

  * •

f-vectors, h-vectors

  * •

flag f-vectors, flag h-vectors

  * •

ab-indices and cd-indices

####  9.0.4. Other topics

  * •

Loday-Ronco Hopf algebra of planar binary trees

  * •

Poirier-Reutenauer Hopf algebra of tableaux

  * •

Reading Hopf algebra of Baxter permutations

  * •

Hopf monoids, e.g. of Hopf algebra of generalized permutohedra, of matroids,
of graphs, Stanley chromatic symmetric functions and Tutte polynomials

  * •

Lam-Pylyavskyy Hopf algebra of set-valued tableaux

  * •

Connes-Kreimer Hopf algebra and renormalization

  * •

Noncommutative symmetric functions and $\Omega\Sigma{\mathbb{C}}P^{\infty}$

  * •

Maschke’s theorem and “integrals” for Hopf algebras

  * •

Nichols-Zoeller structure theorem and group-like elements

  * •

Cartier-Milnor-Moore structure theorem and primitive elements

  * •

Quasi-triangular Hopf algebras and quantum groups

  * •

The Steenrod algebra, its dual, and tree Hopf algebras

  * •

Ringel-Hall algebras of quivers

  * •

Ellis-Khovanov odd symmetric function Hopf algebras (see also Lauda-Russell)

Student talks given in class were:

  1. (1)

Al Garver, on Maschke’s theorem for finite-dimensional Hopf algebras

  2. (2)

Jonathan Hahn, on the paper by Humpert and Martin.

  3. (3)

Emily Gunawan, on the paper by Lam, Lauve and Sottile.

  4. (4)

Jonas Karlsson, on the paper by Connes and Kreimer

  5. (5)

Thomas McConville, on Butcher’s group and generalized Runge-Kutta methods.

  6. (6)

Cihan Bahran, on universal enveloping algebras and the Poincaré-Birkhoff-Witt
theorem.

  7. (7)

Theodosios Douvropolos, on the Cartier-Milnor-Moore theorem.

  8. (8)

Alex Csar, on the Loday-Ronco Hopf algebra of binary trees

  9. (9)

Kevin Dilks, on Reading’s Hopf algebra of (twisted) Baxter permutations

  10. (10)

Becky Patrias, on the paper by Lam and Pylyavskyy

  11. (11)

Meng Wu, on multiple zeta values and Hoffman’s homomorphism from
$\operatorname{QSym}$

##  10\. Some open problems and conjectures

  * •

Is there a proof of the Assaf-McNamara skew Pieri rule that gives a resolution
of Specht or Schur/Weyl modules whose character corresponds to
$s_{\lambda/\mu}h_{n}$, whose terms model their alternating sum?

  * •

Explicit antipodes in the Lam-Pylyavskyy Hopf algebras? (Answered by Patrias
in [136].)

  * •

P. McNamara’s question [123, Question 7.1]: are $P$-partition enumerators
irreducible for connected posets $P$?

  * •

Stanley’s question: are the only $P$-partition enumerators which are symmetric
(not just quasisymmetric) those for which $P$ is a skew-shape with a column-
strict labelling?

  * •

Does Stanley’s chromatic symmetric function distinguish trees?

  * •

Hoffman’s stuffle conjecture

  * •

Billera-Brenti’s nonnegativity conjecture for the total $cd$-index of Bruhat
intervals ([19, Conjecture 6.1])

##  11\. Appendix: Some basics

In this appendix, we briefly discuss some basic notions from linear algebra
and elementary combinatorics that are used in these notes.

###  11.1. Linear expansions and triangularity

In this Section, we shall recall some fundamental results from linear algebra
(most importantly, the notions of a change-of-basis matrix and of a
unitriangular matrix), but in greater generality than how it is usually done
in textbooks. We shall use these results later when studying bases of
combinatorial Hopf algebras; but per se, this section has nothing to do with
Hopf algebras.

####  11.1.1. Matrices

Let us first define the notion of a matrix whose rows and columns are indexed
by arbitrary objects (as opposed to numbers):285285285As before, $\mathbf{k}$
denotes a commutative ring.

######  Definition 11.1.

Let $S$ and $T$ be two sets. An $S\times T$-matrix over $\mathbf{k}$ shall
mean a family $\left(a_{s,t}\right)_{\left(s,t\right)\in S\times
T}\in\mathbf{k}^{S\times T}$ of elements of $\mathbf{k}$ indexed by elements
of $S\times T$. Thus, the set of all $S\times T$-matrices over $\mathbf{k}$ is
$\mathbf{k}^{S\times T}$.

We shall abbreviate “$S\times T$-matrix over $\mathbf{k}$” by “$S\times
T$-matrix” when the value of $\mathbf{k}$ is clear from the context.

This definition of $S\times T$-matrices generalizes the usual notion of
matrices (i.e., the notion of $n\times m$-matrices): Namely, if
$n\in\mathbb{N}$ and $m\in\mathbb{N}$, then the
$\left\\{1,2,\ldots,n\right\\}\times\left\\{1,2,\ldots,m\right\\}$-matrices
are precisely the $n\times m$-matrices (in the usual meaning of this word). We
shall often use the word “matrix” for both the usual notion of matrices and
for the more general notion of $S\times T$-matrices.

Various concepts defined for $n\times m$-matrices (such as addition and
multiplication of matrices, or the notion of a row) can be generalized to
$S\times T$-matrices in a straightforward way. The following four definitions
are examples of such generalizations:

######  Definition 11.2.

Let $S$ and $T$ be two sets.

  1. (a)

The sum of two $S\times T$-matrices is defined by
$\left(a_{s,t}\right)_{\left(s,t\right)\in S\times
T}+\left(b_{s,t}\right)_{\left(s,t\right)\in S\times
T}=\left(a_{s,t}+b_{s,t}\right)_{\left(s,t\right)\in S\times T}$.

  2. (b)

If $u\in\mathbf{k}$ and if $\left(a_{s,t}\right)_{\left(s,t\right)\in S\times
T}\in\mathbf{k}^{S\times T}$, then we define
$u\left(a_{s,t}\right)_{\left(s,t\right)\in S\times T}$ to be the $S\times
T$-matrix $\left(ua_{s,t}\right)_{\left(s,t\right)\in S\times T}$.

  3. (c)

Let $A=\left(a_{s,t}\right)_{\left(s,t\right)\in S\times T}$ be an $S\times
T$-matrix. For every $s\in S$, we define the $s$-th row of $A$ to be the
$\left\\{1\right\\}\times T$-matrix
$\left(a_{s,t}\right)_{\left(i,t\right)\in\left\\{1\right\\}\times T}$.
(Notice that $\left\\{1\right\\}\times T$-matrices are a generalization of row
vectors.) Similarly, for every $t\in T$, we define the $t$-th column of $A$ to
be the $S\times\left\\{1\right\\}$-matrix
$\left(a_{s,t}\right)_{\left(s,i\right)\in S\times\left\\{1\right\\}}$.

######  Definition 11.3.

Let $S$ be a set.

  1. (a)

The $S\times S$ identity matrix is defined to be the $S\times S$-matrix
$\left(\delta_{s,t}\right)_{\left(s,t\right)\in S\times S}$. This $S\times
S$-matrix is denoted by $I_{S}$. (Notice that the $n\times n$ identity matrix
$I_{n}$ is $I_{\left\\{1,2,\ldots,n\right\\}}$ for each $n\in\mathbb{N}$.)

  2. (b)

An $S\times S$-matrix $\left(a_{s,t}\right)_{\left(s,t\right)\in S\times S}$
is said to be diagonal if every $\left(s,t\right)\in S\times T$ satisfying
$s\neq t$ satisfies $a_{s,t}=0$.

  3. (c)

Let $A=\left(a_{s,t}\right)_{\left(s,t\right)\in S\times S}$ be an $S\times
S$-matrix. The diagonal of $A$ means the family $\left(a_{s,s}\right)_{s\in
S}$. The diagonal entries of $A$ are the entries of this diagonal
$\left(a_{s,s}\right)_{s\in S}$.

######  Definition 11.4.

Let $S$, $T$ and $U$ be three sets. Let
$A=\left(a_{s,t}\right)_{\left(s,t\right)\in S\times T}$ be an $S\times
T$-matrix, and let $B=\left(b_{t,u}\right)_{\left(t,u\right)\in T\times U}$ be
a $T\times U$-matrix. Assume that the sum $\sum_{t\in T}a_{s,t}b_{t,u}$ is
well-defined for every $\left(s,u\right)\in S\times U$. (For example, this is
guaranteed to hold if the set $T$ is finite. For infinite $T$, it may and may
not hold.) Then, the $S\times U$-matrix $AB$ is defined by

$AB=\left(\sum_{t\in T}a_{s,t}b_{t,u}\right)_{\left(s,u\right)\in S\times U}.$

######  Definition 11.5.

Let $S$ and $T$ be two finite sets. We say that an $S\times T$-matrix $A$ is
invertible if and only if there exists a $T\times S$-matrix $B$ satisfying
$AB=I_{S}$ and $BA=I_{T}$. In this case, this matrix $B$ is unique; it is
denoted by $A^{-1}$ and is called the inverse of $A$.

The definitions that we have just given are straightforward generalizations of
the analogous definitions for $n\times m$-matrices; thus, unsurprisingly, many
properties of $n\times m$-matrices still hold for $S\times T$-matrices. For
example:

######  Proposition 11.6.

  1. (a)

Let $S$ and $T$ be two sets. Let $A$ be an $S\times T$-matrix. Then,
$I_{S}A=A$ and $AI_{T}=A$.

  2. (b)

Let $S$, $T$ and $U$ be three sets such that $T$ is finite. Let $A$ and $B$ be
two $S\times T$-matrices. Let $C$ be a $T\times U$-matrix. Then,
$\left(A+B\right)C=AC+BC$.

  3. (c)

Let $S$, $T$, $U$ and $V$ be four sets such that $T$ and $U$ are finite. Let
$A$ be an $S\times T$-matrix. Let $B$ be a $T\times U$-matrix. Let $C$ be a
$U\times V$-matrix. Then, $\left(AB\right)C=A\left(BC\right)$.

The proof of Proposition 11.6 (and of similar properties that will be left
unstated) is analogous to the proofs of the corresponding properties of
$n\times m$-matrices.286286286A little warning: In Proposition 11.6(c), the
condition that $T$ and $U$ be finite can be loosened (we leave this to the
interested reader), but cannot be completely disposed of. It can happen that
both $\left(AB\right)C$ and $A\left(BC\right)$ are defined, but
$\left(AB\right)C=A\left(BC\right)$ does not hold (if we remove this
condition). For example, this happens if $S=\mathbb{Z}$, $T=\mathbb{Z}$,
$U=\mathbb{Z}$, $V=\mathbb{Z}$, $A=\left(\begin{cases}1,&\text{if }i\geq j;\\\
0,&\text{if
}i<j\end{cases}\right)_{\left(i,j\right)\in\mathbb{Z}\times\mathbb{Z}}$,
$B=\left(\delta_{i,j}-\delta_{i,j+1}\right)_{\left(i,j\right)\in\mathbb{Z}\times\mathbb{Z}}$
and $C=\left(\begin{cases}0,&\text{if }i\geq j;\\\ 1,&\text{if
}i<j\end{cases}\right)_{\left(i,j\right)\in\mathbb{Z}\times\mathbb{Z}}$.
(Indeed, in this example, it is easy to check that $AB=I_{\mathbb{Z}}$ and
$BC=-I_{\mathbb{Z}}$ and thus
$\underbrace{\left(AB\right)}_{=I_{\mathbb{Z}}}C=I_{\mathbb{Z}}C=C\neq-A=A\underbrace{\left(-I_{\mathbb{Z}}\right)}_{=BC}=A\left(BC\right)$.)
This seeming paradox is due to the subtleties of rearranging infinite sums
(similarly to how a conditionally convergent series of real numbers can change
its value when its entries are rearranged). As a consequence of these
properties, it is easy to see that if $S$ is any finite set, then
$\mathbf{k}^{S\times S}$ is a $\mathbf{k}$-algebra.

In general, $S\times T$-matrices (unlike $n\times m$-matrices) do not have a
predefined order on their rows and their columns. Thus, the classical notion
of a triangular $n\times n$-matrix cannot be generalized to a notion of a
“triangular $S\times S$-matrix” when $S$ is just a set with no additional
structure. However, when $S$ is a poset, such a generalization can be made:

######  Definition 11.7.

Let $S$ be a poset. Let $A=\left(a_{s,t}\right)_{\left(s,t\right)\in S\times
S}$ be an $S\times S$-matrix.

  1. (a)

The matrix $A$ is said to be triangular if and only if every
$\left(s,t\right)\in S\times S$ which does not satisfy $t\leq s$ must satisfy
$a_{s,t}=0$. (Here, $\leq$ denotes the smaller-or-equal relation of the poset
$S$.)

  2. (b)

The matrix $A$ is said to be unitriangular if and only if $A$ is triangular
and has the further property that, for every $s\in S$, we have $a_{s,s}=1$.

  3. (c)

The matrix $A$ is said to be invertibly triangular if and only if $A$ is
triangular and has the further property that, for every $s\in S$, the element
$a_{s,s}$ of $\mathbf{k}$ is invertible.

Of course, all three notions of “triangular”, “unitriangular” and “invertibly
triangular” depend on the partial order on $S$.

Clearly, every invertibly triangular $S\times S$-matrix is triangular. Also,
every unitriangular $S\times S$-matrix is invertibly triangular (because the
element $1$ of $\mathbf{k}$ is invertible).

We can restate the definition of “invertibly triangular” as follows: The
matrix $A$ is said to be invertibly triangular if and only if it is triangular
and its diagonal entries are invertible. Similarly, we can restate the
definition of “unitriangular” as follows: The matrix $A$ is said to be
unitriangular if and only if it is triangular and all its diagonal entries
equal $1$.

Definition 11.7(a) generalizes both the notion of upper-triangular matrices
and the notion of lower-triangular matrices. To wit:

######  Example 11.8.

Let $n\in\mathbb{N}$. Let $N_{1}$ be the poset whose ground set is
$\left\\{1,2,\ldots,n\right\\}$ and whose smaller-or-equal relation $\leq_{1}$
is given by

$s\leq_{1}t\ \Longleftrightarrow\ s\leq t\text{ (as integers).}$

(This is the usual order relation on this set.) Let $N_{2}$ be the poset whose
ground set is $\left\\{1,2,\ldots,n\right\\}$ and whose order relation
$\leq_{2}$ is given by

$s\leq_{2}t\ \Longleftrightarrow\ s\geq t\text{ (as integers).}$

Let $A\in\mathbf{k}^{n\times n}$.

  1. (a)

The matrix $A$ is upper-triangular if and only if $A$ is triangular when
regarded as an $N_{1}\times N_{1}$-matrix.

  2. (b)

The matrix $A$ is lower-triangular if and only if $A$ is triangular when
regarded as an $N_{2}\times N_{2}$-matrix.

More interesting examples of triangular matrices are obtained when the order
on $S$ is not a total order:

######  Example 11.9.

Let $S$ be the poset whose ground set is $\left\\{1,2,3\right\\}$ and whose
smaller relation $<_{S}$ is given by $1<_{S}2$ and $3<_{S}2$. Then, the
triangular $S\times S$-matrices are precisely the $3\times 3$-matrices of the
form $\left(\begin{array}[c]{ccc}a_{1,1}&0&0\\\ a_{2,1}&a_{2,2}&a_{2,3}\\\
0&0&a_{3,3}\end{array}\right)$ with
$a_{1,1},a_{2,1},a_{2,2},a_{2,3},a_{3,3}\in\mathbf{k}$.

We shall now state some basic properties of triangular matrices:

######  Proposition 11.10.

Let $S$ be a finite poset.

  1. (a)

The triangular $S\times S$-matrices form a subalgebra of the
$\mathbf{k}$-algebra $\mathbf{k}^{S\times S}$.

  2. (b)

The invertibly triangular $S\times S$-matrices form a group with respect to
multiplication.

  3. (c)

The unitriangular $S\times S$-matrices form a group with respect to
multiplication.

  4. (d)

Any invertibly triangular $S\times S$-matrix is invertible, and its inverse is
again invertibly triangular.

  5. (e)

Any unitriangular $S\times S$-matrix is invertible, and its inverse is again
unitriangular.

######  Exercise 11.11.

Prove Proposition 11.10.

####  11.1.2. Expansion of a family in another

We will often study situations where two families $\left(e_{s}\right)_{s\in
S}$ and $\left(f_{t}\right)_{t\in T}$ of vectors in a $\mathbf{k}$-module $M$
are given, and the vectors $e_{s}$ can be written as linear combinations of
the vectors $f_{t}$. In such situations, we can form an $S\times T$-matrix out
of the coefficients of these linear combinations; this is one of the ways how
matrices arise in the theory of modules. Let us define the notations we are
going to use in such situations:

######  Definition 11.12.

Let $M$ be a $\mathbf{k}$-module. Let $\left(e_{s}\right)_{s\in S}$ and
$\left(f_{t}\right)_{t\in T}$ be two families of elements of $M$. (The sets
$S$ and $T$ may and may not be finite.)

Let $A=\left(a_{s,t}\right)_{\left(s,t\right)\in S\times T}$ be an $S\times
T$-matrix. Assume that, for every $s\in S$, all but finitely many $t\in T$
satisfy $a_{s,t}=0$. (This assumption is automatically satisfied if $T$ is
finite.)

We say that the family $\left(e_{s}\right)_{s\in S}$ expands in the family
$\left(f_{t}\right)_{t\in T}$ through the matrix $A$ if

(11.1) $\text{every }s\in S\text{ satisfies }e_{s}=\sum_{t\in T}a_{s,t}f_{t}.$

In this case, we furthermore say that the matrix $A$ is a change-of-basis
matrix (or transition matrix) from the family $\left(e_{s}\right)_{s\in S}$ to
the family $\left(f_{t}\right)_{t\in T}$.

######  Remark 11.13.

The notation in Definition 11.12 is not really standard; even we ourselves
will occasionally deviate in its use. In the formulation “the family
$\left(e_{s}\right)_{s\in S}$ expands in the family $\left(f_{t}\right)_{t\in
T}$ through the matrix $A$”, the word “in” can be replaced by “with respect
to”, and the word “through” can be replaced by “using”.

The notion of a “change-of-basis matrix” is slightly misleading, because
neither of the families $\left(e_{s}\right)_{s\in S}$ and
$\left(f_{t}\right)_{t\in T}$ has to be a basis. Our use of the words
“transition matrix” should not be confused with the different meaning that
these words have in the theory of Markov chains. The indefinite article in “a
change-of-basis matrix” is due to the fact that, for given families
$\left(e_{s}\right)_{s\in S}$ and $\left(f_{t}\right)_{t\in T}$, there might
be more than one change-of-basis matrix from $\left(e_{s}\right)_{s\in S}$ to
$\left(f_{t}\right)_{t\in T}$. (There also might be no such matrix.) When
$\left(e_{s}\right)_{s\in S}$ and $\left(f_{t}\right)_{t\in T}$ are bases of
the $\mathbf{k}$-module $M$, there exists precisely one change-of-basis matrix
from $\left(e_{s}\right)_{s\in S}$ to $\left(f_{t}\right)_{t\in T}$.

So a change-of-basis matrix $A=\left(a_{s,t}\right)_{\left(s,t\right)\in
S\times T}$ from one family $\left(e_{s}\right)_{s\in S}$ to another family
$\left(f_{t}\right)_{t\in T}$ allows us to write the elements of the former
family as linear combinations of the elements of the latter (using (11.1)).
When such a matrix $A$ is invertible (and the sets $S$ and $T$ are
finite287287287We are requiring the finiteness of $S$ and $T$ mainly for the
sake of simplicity. We could allow $S$ and $T$ to be infinite, but then we
would have to make some finiteness requirements on $A$ and $A^{-1}$.), it also
(indirectly) allows us to do the opposite: i.e., to write the elements of the
latter family as linear combinations of the elements of the former. This is
because if $A$ is an invertible change-of-basis matrix from
$\left(e_{s}\right)_{s\in S}$ to $\left(f_{t}\right)_{t\in T}$, then $A^{-1}$
is a change-of-basis matrix from $\left(f_{t}\right)_{t\in T}$ to
$\left(e_{s}\right)_{s\in S}$. This is part (a) of the following theorem:

######  Theorem 11.14.

Let $M$ be a $\mathbf{k}$-module. Let $S$ and $T$ be two finite sets. Let
$\left(e_{s}\right)_{s\in S}$ and $\left(f_{t}\right)_{t\in T}$ be two
families of elements of $M$.

Let $A$ be an invertible $S\times T$-matrix. Thus, $A^{-1}$ is a $T\times
S$-matrix.

Assume that the family $\left(e_{s}\right)_{s\in S}$ expands in the family
$\left(f_{t}\right)_{t\in T}$ through the matrix $A$. Then:

  1. (a)

The family $\left(f_{t}\right)_{t\in T}$ expands in the family
$\left(e_{s}\right)_{s\in S}$ through the matrix $A^{-1}$.

  2. (b)

The $\mathbf{k}$-submodule of $M$ spanned by the family
$\left(e_{s}\right)_{s\in S}$ is the $\mathbf{k}$-submodule of $M$ spanned by
the family $\left(f_{t}\right)_{t\in T}$.

  3. (c)

The family $\left(e_{s}\right)_{s\in S}$ spans the $\mathbf{k}$-module $M$ if
and only if the family $\left(f_{t}\right)_{t\in T}$ spans the
$\mathbf{k}$-module $M$.

  4. (d)

The family $\left(e_{s}\right)_{s\in S}$ is $\mathbf{k}$-linearly independent
if and only if the family $\left(f_{t}\right)_{t\in T}$ is
$\mathbf{k}$-linearly independent.

  5. (e)

The family $\left(e_{s}\right)_{s\in S}$ is a basis of the $\mathbf{k}$-module
$M$ if and only if the family $\left(f_{t}\right)_{t\in T}$ is a basis of the
$\mathbf{k}$-module $M$.

######  Exercise 11.15.

Prove Theorem 11.14.

######  Definition 11.16.

Let $M$ be a $\mathbf{k}$-module. Let $S$ be a finite poset. Let
$\left(e_{s}\right)_{s\in S}$ and $\left(f_{s}\right)_{s\in S}$ be two
families of elements of $M$.

  1. (a)

We say that the family $\left(e_{s}\right)_{s\in S}$ expands triangularly in
the family $\left(f_{s}\right)_{s\in S}$ if and only if there exists a
triangular $S\times S$-matrix $A$ such that the family
$\left(e_{s}\right)_{s\in S}$ expands in the family $\left(f_{s}\right)_{s\in
S}$ through the matrix $A$.

  2. (b)

We say that the family $\left(e_{s}\right)_{s\in S}$ expands invertibly
triangularly in the family $\left(f_{s}\right)_{s\in S}$ if and only if there
exists an invertibly triangular $S\times S$-matrix $A$ such that the family
$\left(e_{s}\right)_{s\in S}$ expands in the family $\left(f_{s}\right)_{s\in
S}$ through the matrix $A$.

  3. (c)

We say that the family $\left(e_{s}\right)_{s\in S}$ expands unitriangularly
in the family $\left(f_{s}\right)_{s\in S}$ if and only if there exists a
unitriangular $S\times S$-matrix $A$ such that the family
$\left(e_{s}\right)_{s\in S}$ expands in the family $\left(f_{s}\right)_{s\in
S}$ through the matrix $A$.

Clearly, if the family $\left(e_{s}\right)_{s\in S}$ expands unitriangularly
in the family $\left(f_{s}\right)_{s\in S}$, then it also expands invertibly
triangularly in the family $\left(f_{s}\right)_{s\in S}$ (because any
unitriangular matrix is an invertibly triangular matrix).

We notice that in Definition 11.16, the two families $\left(e_{s}\right)_{s\in
S}$ and $\left(f_{s}\right)_{s\in S}$ must be indexed by one and the same set
$S$.

The concepts of “expanding triangularly”, “expanding invertibly triangularly”
and “expanding unitriangularly” can also be characterized without referring to
matrices, as follows:

######  Remark 11.17.

Let $M$ be a $\mathbf{k}$-module. Let $S$ be a finite poset. Let
$\left(e_{s}\right)_{s\in S}$ and $\left(f_{s}\right)_{s\in S}$ be two
families of elements of $M$. Let $<$ denote the smaller relation of the poset
$S$, and let $\leq$ denote the smaller-or-equal relation of the poset $S$.
Then:

  1. (a)

The family $\left(e_{s}\right)_{s\in S}$ expands triangularly in the family
$\left(f_{s}\right)_{s\in S}$ if and only if every $s\in S$ satisfies

$e_{s}=\left(\text{a }\mathbf{k}\text{-linear combination of the elements
}f_{t}\text{ for }t\in S\text{ satisfying }t\leq s\right).$

  2. (b)

The family $\left(e_{s}\right)_{s\in S}$ expands invertibly triangularly in
the family $\left(f_{s}\right)_{s\in S}$ if and only if every $s\in S$
satisfies

$e_{s}=\alpha_{s}f_{s}+\left(\text{a }\mathbf{k}\text{-linear combination of
the elements }f_{t}\text{ for }t\in S\text{ satisfying }t<s\right)$

for some invertible $\alpha_{s}\in\mathbf{k}$.

  3. (c)

The family $\left(e_{s}\right)_{s\in S}$ expands unitriangularly in the family
$\left(f_{s}\right)_{s\in S}$ if and only if every $s\in S$ satisfies

$e_{s}=f_{s}+\left(\text{a }\mathbf{k}\text{-linear combination of the
elements }f_{t}\text{ for }t\in S\text{ satisfying }t<s\right).$

All three parts of Remark 11.17 follow easily from the definitions.

######  Example 11.18.

Let $n\in\mathbb{N}$. For this example, let $S$ be the poset
$\left\\{1,2,\ldots,n\right\\}$ (with its usual order). Let $M$ be a
$\mathbf{k}$-module, and let $\left(e_{s}\right)_{s\in S}$ and
$\left(f_{s}\right)_{s\in S}$ be two families of elements of $M$. We shall
identify these families $\left(e_{s}\right)_{s\in S}$ and
$\left(f_{s}\right)_{s\in S}$ with the $n$-tuples
$\left(e_{1},e_{2},\ldots,e_{n}\right)$ and
$\left(f_{1},f_{2},\ldots,f_{n}\right)$. Then, the family
$\left(e_{s}\right)_{s\in S}=\left(e_{1},e_{2},\ldots,e_{n}\right)$ expands
triangularly in the family $\left(f_{s}\right)_{s\in
S}=\left(f_{1},f_{2},\ldots,f_{n}\right)$ if and only if, for every
$s\in\left\\{1,2,\ldots,n\right\\}$, the vector $e_{s}$ is a
$\mathbf{k}$-linear combination of $f_{1},f_{2},\ldots,f_{s}$. Moreover, the
family $\left(e_{s}\right)_{s\in S}=\left(e_{1},e_{2},\ldots,e_{n}\right)$
expands unitriangularly in the family $\left(f_{s}\right)_{s\in
S}=\left(f_{1},f_{2},\ldots,f_{n}\right)$ if and only if, for every
$s\in\left\\{1,2,\ldots,n\right\\}$, the vector $e_{s}$ is a sum of $f_{s}$
with a $\mathbf{k}$-linear combination of $f_{1},f_{2},\ldots,f_{s-1}$.

######  Corollary 11.19.

Let $M$ be a $\mathbf{k}$-module. Let $S$ be a finite poset. Let
$\left(e_{s}\right)_{s\in S}$ and $\left(f_{s}\right)_{s\in S}$ be two
families of elements of $M$. Assume that the family $\left(e_{s}\right)_{s\in
S}$ expands invertibly triangularly in the family $\left(f_{s}\right)_{s\in
S}$. Then:

  1. (a)

The family $\left(f_{s}\right)_{s\in S}$ expands invertibly triangularly in
the family $\left(e_{s}\right)_{s\in S}$.

  2. (b)

The $\mathbf{k}$-submodule of $M$ spanned by the family
$\left(e_{s}\right)_{s\in S}$ is the $\mathbf{k}$-submodule of $M$ spanned by
the family $\left(f_{s}\right)_{s\in S}$.

  3. (c)

The family $\left(e_{s}\right)_{s\in S}$ spans the $\mathbf{k}$-module $M$ if
and only if the family $\left(f_{s}\right)_{s\in S}$ spans the
$\mathbf{k}$-module $M$.

  4. (d)

The family $\left(e_{s}\right)_{s\in S}$ is $\mathbf{k}$-linearly independent
if and only if the family $\left(f_{s}\right)_{s\in S}$ is
$\mathbf{k}$-linearly independent.

  5. (e)

The family $\left(e_{s}\right)_{s\in S}$ is a basis of the $\mathbf{k}$-module
$M$ if and only if the family $\left(f_{s}\right)_{s\in S}$ is a basis of the
$\mathbf{k}$-module $M$.

######  Exercise 11.20.

Prove Remark 11.17 and Corollary 11.19.

An analogue of Corollary 11.19 can be stated for unitriangular expansions, but
we leave this to the reader.

## Acknowledgements

The authors thank the following for helpful comments and/or teaching them
about Hopf algebras: Marcelo Aguiar, Federico Ardila, Lou Billera, Richard
Ehrenborg, Mark Haiman, Florent Hivert, Christophe Hohlweg, Jia Huang, Jang
Soo Kim, Aaron Lauve, Dominique Manchon, John Palmieri, Alexander Postnikov,
Margie Readdy, Nathan Reading, Hans-Jürgen Schneider, Richard Stanley, Muge
Taskin, Jean-Yves Thibon.

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