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#Symmetric functions in noncommuting variables

The content on this page is mainly based on [ALW22], and has been contributed by F. Aliniaeifard.

F. Aliniaeifard, S. X. Li, and Stephanie van Willigenburg define Schur functions in noncommuting variables using a noncommutative Jacobi–Trudi determinant [ALW22]. These functions commute to ordinary Schur functions, a subset indexed by set partitions forms a basis of \(\mathrm{NCSym},\) and the construction refines the Rosas–Sagan Schur functions.

The graded Hopf algebra of symmetric functions in noncommuting variables is \[\mathrm{NCSym} = \mathrm{NCSym}^0 \oplus \mathrm{NCSym}^1 \oplus \dotsb \subset \setQ \langle \langle x_1, x_2, \dotsc \rangle\rangle\]. Here \(\langle\langle x_1,x_2,\dotsc\rangle\rangle\) denotes formal power series in noncommuting variables, \(\mathrm{NCSym}^0=\mathrm{span}\{1\},\) 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, are a superset of noncommutative symmetric functions, which are indexed by integer compositions.

The monomial symmetric function in \(\mathrm{NCSym}\) , \(m_\pi,\) is given by \[m_\pi = \sum _{(i_1, i_2, \dotsc, i_n)} x_{i_1}x_{i_2} \dotsm x_{i_n}\] summed over all tuples \((i_1, i_2, \dotsc, i_n)\) with \(i_j=i_k\) if and only if \(j\) and \(k\) are in the same block of \(\pi.\)

Example (Expansion of \(m_{13|2}\)).

\(m_{13|2}=x_1x_2x_1+x_2x_1x_2+x_1x_3x_1+x_3x_1x_3+x_2x_3x_2+x_3x_2x_3+\cdots\)

The power-sum symmetric function in \(\mathrm{NCSym}\) , \(p_\pi,\) is given by \[p_\pi = \sum _{(i_1, i_2, \dotsc, i_n)} x_{i_1}x_{i_2} \dotsm x_{i_n}\] summed over all tuples \((i_1, i_2, \dotsc, i_n)\) with \(i_j=i_k\) if \(j\) and \(k\) are in the same block of \(\pi.\)

Example (Expansion of \(p_{13|2}\)).

\(p_{13|2}=x_1x_2x_1+x_2x_1x_2+\cdots + x_1^3+x_2^3 +\cdots\)

The elementary symmetric function in \(\mathrm{NCSym}\) , \(e_\pi,\) is given by \[e_\pi = \sum _{(i_1, i_2, \dotsc, i_n)} x_{i_1}x_{i_2} \dotsm x_{i_n}\] summed over all tuples \((i_1, i_2, \dotsc, i_n)\) with \(i_j\neq i_k\) if \(j\) and \(k\) are in the same block of \(\pi.\)

Example (Expansion of \(e_{13|2}\)).

\(e_{13|2}= {x_1x_1x_2+x_1x_2x_2+x_2x_2x_1+x_2x_1x_1}+\cdots + x_1x_2x_3+x_2x_3x_4 +\cdots\)

The complete homogeneous symmetric function in \(\mathrm{NCSym}\) , \(h_\pi,\) is given by \[h_{\pi}= \sum_{\eta} \sum_{(i_1,i_2,\dotsc,i_n)} x_{i_{\eta(1)}}x_{i_{\eta(2)}}\dotsm x_{i_{\eta(n)}}\] where the first sum is over all \(\eta\in \symS_n\) that fixes the blocks of \(\pi,\) and the second sum is over all \((i_1,i_2,\dotsc,i_n)\in \setN^n\) such that if \(j\) and \(k\) are in the same block of \(\pi\) with \(j \lt k,\) then \(i_j\leq i_k.\)

Example (Expansion of \(h_{13|2}\)).

\(h_{13|2}= 2 m_{123} + m_{12|3} + m_{1|23} + 2 m_{13|2} + m_{1|2|3}\)

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 \[\delta \circ (x_{i_1}x_{i_2} \dotsm x_{i_n}) = x_{i_{\delta^{-1}(1)}}x_{i_{\delta^{-1}(2)}} \dotsm x_{i_{\delta^{-1}(n)}}\] and extend linearly. In [p. 219, RS06], it is noted that if \(\pi\) is a basis element of any of the above bases of \(\mathrm{NCSym}\) and \(\delta\) is a permutation, then

\[\delta \circ b_\pi = b_{\delta\pi}\] 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 \[\mathbf{det}(A) = \sum_{\varepsilon \in \symS_n} \sign (\varepsilon) a_{1\varepsilon (1)}a_{2\varepsilon (2)} \dotsm a_{n\varepsilon (n)}\] 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, we define the source functions from which they are built.

Definition

Let \(\lambda / \mu\) be a skew diagram. Then the source skew Schur function in noncommuting variables \(s_{[\lambda/\mu]}\) is defined to be \[s_{[\lambda/\mu]} = \mathbf{det} \left( \frac{1}{(\lambda _i -\mu _j - i +j)! } h_{[\lambda _i -\mu_j - i + j]}\right) _{1\leq i,j \leq \ell(\lambda)}\] 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 .

Example

The source Schur function in noncommuting variables \(s_{[21]}\) is \[\begin{aligned} s_{[21]} & = {\mathbf{det} \begin{pmatrix} \frac{1}{2!} h_{[2]}& \frac{1}{3!} h_{[3]}\\ \frac{1}{0!} h_{[0]}& \frac{1}{1!} h_{[1]} \end{pmatrix}} = \mathbf{det} \begin{pmatrix} \frac{1}{2!} h_{12}& \frac{1}{3!} h_{123}\\ \frac{1}{0!} h_{\emptyset}& \frac{1}{1!} h_{1} \end{pmatrix}\\ &= \frac{1}{2!} h_{12} \frac{1}{1!} h_{1} - \frac{1}{3!} h_{123}\frac{1}{0!} h_{\emptyset} = \frac{1}{2} h_{12|3} - \frac{1}{6} h_{123}. \end{aligned}\]

Meanwhile, the source skew Schur function in noncommuting variables \(s_{[22|1]}\) is \[\begin{aligned} s_{[22|1]} &= {\mathbf{det} \begin{pmatrix} \frac{1}{1!} h_{[1]}& \frac{1}{3!} h_{[3]}\\ \frac{1}{0!} h_{[0]}& \frac{1}{2!} h_{[2]} \end{pmatrix}} = \mathbf{det} \begin{pmatrix} \frac{1}{1!} h_{1}& \frac{1}{3!} h_{123}\\ \frac{1}{0!} h_{\emptyset}& \frac{1}{2!} h_{12} \end{pmatrix}\\ &= \frac{1}{1!} h_{1}\frac{1}{2!} h_{12} - \frac{1}{3!} h_{123}\frac{1}{0!} h_{\emptyset} = \frac{1}{2} h_{1|23} - \frac{1}{6} h_{123}. \end{aligned}\]

#Schur functions in noncommuting 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 \([n]\) appear 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, \(T \leftrightarrow (\delta_T, sh(T))\) is a bijection, where \(sh(T)\) is the shape of \(T.\)

Now consider the set of all Young tableaux \(T\) such that

  1. \(sh(T) = \lambda\) for some fixed integer partition \(\lambda\vdash n,\)

  2. the entries in each row of \(T\) increase from left to right,

  3. if \(\lambda = \lambda _1 \lambda _2\dotsm \lambda _{\ell(\lambda)}\) 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 \([n]:\) the Young tableau \(T_\pi\) corresponds to the set partition \(\pi\) if and only if the entries in each row of \(T_\pi\) are precisely the blocks of \(\pi,\) and the integer partition determined by the block sizes of \(\pi\) is \(sh(T_\pi).\) 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.

Example (Obtaining \(\delta_T\) from tableaux).

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 \(\delta_\pi = \delta_{T_\pi} = 169378452.\)

Definition

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 \[\label{eq:skewNCSchur} s_{(\delta, \lambda /\mu)} = \delta \circ s_{[\lambda/\mu]} = \delta \circ \mathbf{det} \left( \frac{1}{(\lambda _i -\mu _j - i +j)! } h_{[\lambda _i -\mu_j - i + j]}\right) _{1\leq i,j \leq \ell(\lambda)}.\] Moreover, if \(\mu = \emptyset,\) then we call \(s_{(\delta, \lambda)}\) a Schur function in noncommuting variables.

Furthermore, if \(\pi \vdash [n]\) and \(\lambda (\pi) = \lambda _1 \lambda _2 \dotsm \lambda _{\ell(\pi)},\) then the standard Schur function in noncommuting variables \(s_\pi\) is defined to be \[s_{\pi} = s_{(\delta _\pi, \lambda (\pi))}= \delta _\pi \circ s_{[\lambda (\pi)]} = \delta_\pi \circ \mathbf{det} \left( \frac{1}{(\lambda _i - i +j)! } h_{[\lambda _i - i + j]}\right) _{1\leq i,j \leq \ell(\lambda(\pi))}.\]

Example (Schur functions in noncommuting variables).

If \(\pi = 12|3,\) then \(\delta _\pi = 123 = \mathrm{id}.\)

Hence, the standard Schur function in noncommuting variables \(s_{12|3}\) is \[\begin{aligned} s_{12|3} &= \mathrm{id} \circ s_{[21]} = \mathrm{id} \circ \mathbf{det} \begin{pmatrix} \frac{1}{2!} h_{12}& \frac{1}{3!} h_{123}\\ \frac{1}{0!} h_{\emptyset}& \frac{1}{1!} h_{1} \end{pmatrix}\\ &= \frac{1}{2!} h_{12} \frac{1}{1!} h_{1} - \frac{1}{3!} h_{123}\frac{1}{0!} h_{\emptyset} = \frac{1}{2} h_{12|3} - \frac{1}{6} h_{123}. \end{aligned}\] If \(\pi = 13|2,\) then \(\delta _\pi = 132.\) Hence, the standard Schur function in noncommuting variables \(s_{13|2}\) is \[s_{13|2} = 132 \circ s_{[21]} = 132\circ \left(\frac{1}{2} h_{12|3} - \frac{1}{6} h_{123}\right) = \frac{1}{2} h_{13|2} - \frac{1}{6} h_{123}.\]

Theorem

The set \(\{s_\pi : \pi\vdash[n],\ n\geq 0\}\) is a basis for \(\mathrm{NCSym}.\)

This basis lives in \(\mathrm{NCSym},\) the algebra of symmetric functions in noncommuting variables. It should be compared with the Schur-like bases of noncommutative symmetric functions \(\mathrm{NSym}:\) the noncommutative Schur functions, the immaculate Schur functions, and the ribbon basis. Under the duality between \(\mathrm{NSym}\) and \(\spaceQSym,\) the immaculate basis is paired with the dual immaculate Schur functions. See [ALW22] for precise connections between Schur functions in noncommuting variables, immaculate Schur functions, and ribbon Schur functions.

Bibliography

  1. [ALW22]Farid Aliniaeifard, Shu Xiao Li and Stephanie Willigenburg. Schur functions in noncommuting variables. Advances in Mathematics, 406:108536, September 2022.
    .bib
    @article{AliniaeifardLiWilligenburg2022,
  doi = {10.1016/j.aim.2022.108536},
  url2 = {https://doi.org/10.1016/j.aim.2022.108536},
  year = {2022},
  month = sep,
  publisher = {Elsevier {BV}},
  volume = {406},
  pages = {108536},
  author = {Farid Aliniaeifard and Shu Xiao Li and Stephanie van Willigenburg},
  title = {Schur functions in noncommuting variables},
  journal = {Advances in Mathematics},
  eprint = {2105.09964}
}
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    @article{GebhardSagan2001,
  doi = {10.1023/a:1011258714032},
  url2 = {https://doi.org/10.1023/a:1011258714032},
  year = {2001},
  title = {A chromatic symmetric function in noncommuting variables},
  publisher = {Springer Science and Business Media {LLC}},
  volume = {13},
  number = {3},
  pages = {227--255},
  author = {David D. Gebhard and Bruce E. Sagan},
  journal = {Journal of Algebraic Combinatorics}
}
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    @article{RosasBruce2006,
 ISSN = {00029947},
 URL = {http://www.jstor.org/stable/3845454},
 author = {Mercedes H. Rosas and Bruce E. Sagan},
 journal = {Transactions of the American Mathematical Society},
 number = {1},
 pages = {215--232},
 publisher = {American Mathematical Society},
 title = {Symmetric Functions in Noncommuting Variables},
 urldate = {2022-10-21},
 volume = {358},
 year = {2006}
}

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