2022-11-08
Symmetric functions in noncommuting variables
The content on this page is mainly based on [ALW22], and has been contributed by Farid Aliniaeifard.
The graded Hopf algebra of symmetric functions in noncommuting variables, $\mathrm{NCSym},$
\[ \mathrm{NCSym} = \mathrm{NCSym}^0 \oplus \mathrm{NCSym}^1 \oplus \cdots \subset \setQ \langle \langle x_1, x_2, \dotsc \rangle\rangle \]where $\langle \langle\cdot \rangle\rangle$ means that the variables do not commute, $\mathrm{NCSym}^0 = \mathrm{span} \{1\}$ and the $n$th graded piece for $n\geq 1$ has the following bases [RS06], known respectively as the $n$th graded piece of the $m$-, $p$-, $e$-, $h$-basis of $\mathrm{NCSym}$,
\[ \mathrm{NCSym}^n = \mathrm{span}\{ m_\pi : \pi\vDash [n]\} = \mathrm{span}\{ p_\pi : \pi\vDash [n]\} = \mathrm{span}\{ e_\pi : \pi\vDash [n]\} = \mathrm{span}\{ {h_\pi} : \pi\vDash [n]\} \]where these functions are defined below, and $\pi$ is a set partition of $[n].$
The symmetric functions in noncommuting variables (indexed by set partitions) is a superset of noncommutative symmetric functions (indexed by integer compositions).
The monomial symmetric function in $\mathrm{NCSym}$, $m_\pi,$ is given by
\[ m_\pi = \sum _{(i_1, i_2, \ldots , i_n)} x_{i_1}x_{i_2} \cdots x_{i_n} \]summed over all tuples $(i_1, i_2, \ldots , i_n)$ with $i_j=i_k$ if and only if $j$ and $k$ are in the same block of $\pi.$
$m_{13|2}=x_1x_2x_1+x_2x_1x_2+x_1x_3x_1+x_3x_1x_3+x_2x_3x_2+x_3x_2x_3+\cdots$
The power sum symmetric function in $\mathrm{NCSym}$, $p_\pi,$ is given by
\[ p_\pi = \sum _{(i_1, i_2, \ldots , i_n)} x_{i_1}x_{i_2} \cdots x_{i_n} \]summed over all tuples $(i_1, i_2, \ldots , i_n)$ with $i_j=i_k$ if $j$ and $k$ are in the same block of $\pi.$
$p_{13|2}=x_1x_2x_1+x_2x_1x_2+\cdots + x_1^3+x_2^3 +\cdots $
The elementary symmetric function in $\mathrm{NCSym}$, $e_\pi,$ is given by
\[ e_\pi = \sum _{(i_1, i_2, \ldots , i_n)} x_{i_1}x_{i_2} \cdots x_{i_n} \]summed over all tuples $(i_1, i_2, \ldots , i_n)$ with $i_j\neq i_k$ if $j$ and $k$ are in the same block of $\pi.$
$e_{13|2}= {x_1x_1x_2+x_1x_2x_2+x_2x_2x_1+x_2x_1x_1}+\cdots + x_1x_2x_3+x_2x_3x_4 +\cdots$
The complete homogeneous symmetric function in $\mathrm{NCSym},$ $h_\pi,$ is given by
\begin{equation} h_{\pi}=\sum_{\eta} \sum_{(i_1,i_2,\ldots,i_n) } x_{i_{\eta(1)}}x_{i_{\eta(2)}}\cdots x_{i_{\eta(n)}} \end{equation}where the first sum is over all $\eta\in \symS_n$ that fixes the blocks of $\pi,$ and, the second sum is over all $(i_1,i_2,\ldots,i_n)\in \setN^n$ such that if $j$ and $k$ are in the same block of $\pi$ with $j \lt k,$ then $i_j\leq i_k.$
$h_{13|2}= 2 m_{123} + m_{12|3} + m_{1|23} + 2 m_{13|2} + m_{1|2|3}$
The permutation map [p. 219, RS06] and [p. 230, GS01], which is an action on places (not variables), is defined as follows. Given $\delta \in \symS_n$ and a monomial of degree $n$ in noncommuting variables, define
\[ \delta \circ (x_{i_1}x_{i_2} \cdots x_{i_n}) = x_{i_{\delta^{-1}(1)}}x_{i_{\delta^{-1}(2)}} \cdots x_{i_{\delta^{-1}(n)}} \]and extend linearly. In [p. 219, RS06] they also noted that if $\pi$ is a basis element of any of the above bases of $\mathrm{NCSym}$ and $\delta$ is a permutation, then
\begin{equation} \delta \circ b_\pi = b_{\delta\pi} \end{equation}where $\delta$ acts on set partitions in the natural way.
The noncommutative analogue of Leibniz' determinantal formula for any matrix $A=(a_{ij}) _{1\leq i,j\leq n}$ with noncommuting entries $a_{ij}$ is defined to be
\begin{equation} \mathbf{det}(A) = \sum_{\varepsilon \in \symS_n} \sign (\varepsilon) a_{1\varepsilon (1)}a_{2\varepsilon (2)}\cdots a_{n\varepsilon (n)} \end{equation}that takes the product of the entries from the top row to the bottom row, and $\sign (\varepsilon)$ is the sign of permutation $\varepsilon.$
Source Schur functions
Before the definition of Schur functions in noncommuting variables, let's define functions that will be their genesis.
Let $\lambda / \mu$ be a skew diagram. Then the source skew Schur function in noncommuting variables $s_{[\lambda/\mu]}$ is defined to be
\begin{equation} s_{[\lambda/\mu]} = \mathbf{det} \left( \frac{1}{(\lambda _i -\mu _j - i +j)! } h_{[\lambda _i -\mu_j - i + j]}\right) _{1\leq i,j \leq \ell(\lambda)} \end{equation}where we set $\mu_j = 0$ for all $\ell(\mu) \lt j \leq \ell(\lambda),$ $h_{[0]}=h_\emptyset = 1$ and any function with a negative index equals 0. When $\mu = \emptyset,$ we call $s_{[\lambda]}$ a source Schur function in noncommuting variables.
The source Schur function in noncommuting variables $s_{[21]}$ is
\begin{align*}s_{[21]} & = {\mathbf{det} \begin{pmatrix} \frac{1}{2!} h_{[2]}& \frac{1}{3!} h_{[3]}\\ \frac{1}{0!} h_{[0]}& \frac{1}{1!} h_{[1]} \end{pmatrix}} = \mathbf{det} \begin{pmatrix} \frac{1}{2!} h_{12}& \frac{1}{3!} h_{123}\\ \frac{1}{0!} h_{\emptyset}& \frac{1}{1!} h_{1} \end{pmatrix}\\ &= \frac{1}{2!} h_{12} \frac{1}{1!} h_{1} - \frac{1}{3!} h_{123}\frac{1}{0!} h_{\emptyset} = \frac{1}{2} h_{12|3} - \frac{1}{6} h_{123}. \end{align*}Meanwhile, the source skew Schur function in noncommuting variables $s_{[22|1]}$ is
\begin{align*}s_{[22|1]} &= {\mathbf{det} \begin{pmatrix} \frac{1}{1!} h_{[1]}& \frac{1}{3!} h_{[3]}\\ \frac{1}{0!} h_{[0]}& \frac{1}{2!} h_{[2]} \end{pmatrix}} = \mathbf{det} \begin{pmatrix} \frac{1}{1!} h_{1}& \frac{1}{3!} h_{123}\\ \frac{1}{0!} h_{\emptyset}& \frac{1}{2!} h_{12} \end{pmatrix}\\ &= \frac{1}{1!} h_{1}\frac{1}{2!} h_{12} - \frac{1}{3!} h_{123}\frac{1}{0!} h_{\emptyset} = \frac{1}{2} h_{1|23} - \frac{1}{6} h_{123}. \end{align*}Schur functions in non-commuting variables
The standard and permuted bases
Recall the definition of skew diagrams and standard Young tableaux. In particular, the permutation $\delta_T \in \symS_n$ of a tableau $T$ is obtained by concatenating the rows of $T,$ from first to last row. This definition makes sense as long as entries from $[n]$ appears exactly once; the tableau is not required for this definition to make sense. Observe that this is different from the reading word of $T$, where rows are read in a different order! Evidently, we have that $T \leftrightarrow (\delta_T, sh(T))$ is a bijection, where $sh(T)$ is the shape of $T.$
Now consider the set of all Young tableaux $T$ such that
- $sh(T) = \lambda$ for some fixed integer partition $\lambda\vdash n,$
- the entries in each row of $T$ increase from left to right,
- if $\lambda = \lambda _1 \lambda _2\cdots \lambda _{\ell(\lambda)}$ and $\lambda _i = \lambda _j$ with $i\lt j,$ then in $T$ \[ \text{(the first entry of row $i$)} \lt \text{(the first entry of row $j$)}. \]
Observe that this set is in bijection with the set consisting of all set partitions $\pi$ of $\vDash [n]$: the Young tableau $T_\pi$ corresponds to set partition $\pi$ if and only if the set of entries for each row of $T_\pi$ are precisely the blocks of $\pi,$ and that the integer partition determined by the block sizes of $\pi$ are given by $sh(T_\pi).$ In this case, define the permutation $\delta_\pi \coloneqq \delta_{T_\pi}.$ Informally, $\delta_\pi$ is obtained from the set-partition $\pi$ by sorting blocks by length decreasingly; blocks with smaller first entry are placed first. Finally, bars separating blocks are erased.
If $T$ is
| $3$ | $8$ | $7$ | ||
| $2$ | ||||
| $1$ | $9$ | $6$ | ||
| $5$ | $4$ |
then $\delta_T = 387219654.$ If $T_\pi$ is
| $1$ | $6$ | $9$ |
| $3$ | $7$ | $8$ |
| $4$ | $5$ | |
| $2$ |
then $\pi = 169|378|45|2 = 169|2|378|45$ and $\delta_\pi = \delta_{T_\pi} = 169378452.$
Let $\lambda/\mu$ be a skew diagram of size $n$ and $\delta \in \symS_n.$ Then the skew Schur function in noncommuting variables $s_{(\delta, \lambda /\mu)}$ is defined to be
\begin{equation}\label{eq:skewNCSchur} s_{(\delta, \lambda /\mu)} = \delta \circ s_{[\lambda/\mu]} = \delta \circ \mathbf{det} \left( \frac{1}{(\lambda _i -\mu _j - i +j)! } h_{[\lambda _i -\mu_j - i + j]}\right) _{1\leq i,j \leq \ell(\lambda)}. \end{equation}Moreover, if $\mu = \emptyset,$ then we call $s_{(\delta, \lambda)}$ a Schur function in noncommuting variables.
Furthermore, if $\pi \vDash [n]$ and $\lambda (\pi) = \lambda _1 \lambda _2 \cdots \lambda _{\ell(\pi)},$ then the standard Schur function in noncommuting variables $s_\pi$ is defined to be
\begin{equation} s_{\pi} = s_{(\delta _\pi, \lambda (\pi))}= \delta _\pi \circ s_{[\lambda (\pi)]} = \delta_\pi \circ \mathbf{det} \left( \frac{1}{(\lambda _i - i +j)! } h_{[\lambda _i - i + j]}\right) _{1\leq i,j \leq \ell(\lambda(\pi))}. \end{equation}If $\pi = 12|3,$ then $\delta _\pi = 123 = \mathrm{id}.$
Hence, the standard Schur function in noncommuting variables $s_{12|3}$ is
\begin{align*} s_{12|3} &= \mathrm{id} \circ s_{[21]} = \mathrm{id} \circ \mathbf{det} \begin{pmatrix} \frac{1}{2!} h_{12}& \frac{1}{3!} h_{123}\\ \frac{1}{0!} h_{\emptyset}& \frac{1}{1!} h_{1} \end{pmatrix}\\ &= \frac{1}{2!} h_{12} \frac{1}{1!} h_{1} - \frac{1}{3!} h_{123}\frac{1}{0!} h_{\emptyset} = \frac{1}{2} h_{12|3} - \frac{1}{6} h_{123}. \end{align*}If $\pi = 13|2,$ then $\delta _\pi = 132.$ Hence, the standard Schur function in noncommuting variables $s_{13|2}$ is
\[ s_{13|2} = 132 \circ s_{[21]} = 132\circ \left(\frac{1}{2} h_{12|3} - \frac{1}{6} h_{123}\right) = \frac{1}{2} h_{13|2} - \frac{1}{6} h_{123}. \]The set $\{ s_\pi \} _{\pi \vDash [n], {n\geq 0}}$ is a basis for $\mathrm{NCSym}.$
See [ALW22] for connections with the immaculate Schur functions, and ribbon Schur functions.
References
- [ALW22] Farid Aliniaeifard, Shu Xiao Li and Stephanie van Willigenburg. Schur functions in noncommuting variables. Advances in Mathematics, 406:108536, September 2022.
- [GS01] David D. Gebhard and Bruce E. Sagan. A chromatic symmetric function in noncommuting variables. Journal of Algebraic Combinatorics, 13(3):227–255, 2001.
- [RS06] Mercedes H. Rosas and Bruce E. Sagan. Symmetric functions in noncommuting variables. Transactions of the American Mathematical Society, 358(1):215–232, 2006.