The symmetric functions catalog

An overview of symmetric functions and related topics

2023-12-05

Shifted Schur polynomials

Shifted Schur functions were studied in [OO96], and are closely related to the factorial Schur polynomials. The shifted Schur functions are specializations of the shifted Jack functions.

Define the falling factorial as

\[ \fallingFactorial{x}{k} \coloneqq x(x-1)\dotsm (x-k+1), \qquad \fallingFactorial{x}{0} \coloneqq 1. \]

The shifted Schur polynomials are defined as the quotient

\[ \schurShifted_\mu(x_1,\dotsc,x_n) \coloneqq \frac{ \det( \fallingFactorial{x_i+n-i}{\mu_j+n-j} ) }{ \det( \fallingFactorial{x_i+n-i}{n-j} ) }. \]

Properties

The shifted Schur polynomials are stable in the sense that

\[ \schurShifted_\mu(x_1,\dotsc,x_n) = \schurShifted_\mu(x_1,\dotsc,x_n,0). \]

Let $H(\mu)$ be the product of hook lengths in the diagram $\mu.$ The shifted Schur functions $\schurShifted_\mu(\xvec)$ is the unique family of shifted symmetric functions with degree $\leq |\mu|$ such that

\[ \schurShifted_\mu(\lambda) = \delta_{\lambda\mu} H(\mu) \]

for all $\lambda$ such that $|\lambda| \leq |\mu|.$ This is commonly referred to as the vanishing property.

Let $l=|\lambda|$ and $k=|\mu|.$ Then

\[ \frac{f^{\lambda/\mu}}{f^{\lambda}} = \frac{\schurShifted_\mu(\lambda)}{\fallingFactorial{l}{k}} \]

where $f^{\lambda/\mu}$ is the number of skew standard Young tableaux of shape $\lambda/\mu.$

Combinatorial formula

There is a combinatorial formula, see [OO96],

\[ \schurShifted_\mu(\xvec) = \sum_{T \in \mathrm{RSSYT}(\mu)} \prod_{s \in \mu} (x_{T(s)} - c(s)) \]

where $\mathrm{RSSYT}(\mu)$ is the set of reverse semi-standard Young tableaux. These are fillings of $\mu$ with weakly decreasing rows and strictly decreasing columns. Here, $c(s)$ is the content of the square $s.$

Note that this formula implies that

\[ \schurShifted_\mu(\xvec) = \schurS_\mu(\xvec) + \text{ lower order terms}. \]

Jacobi–Trudi identities

Define the shifted elementary and complete homogeneous symmetric functions as

\[ \elementaryE^*_r(\xvec) = \schurShifted_{(1^r)}(\xvec)\qquad \completeH^*_r(\xvec) = \schurShifted_{(r)}(\xvec). \]

Let $H^*(u)$ be the formal power series in $u^{-1}$:

\[ H^*(u) \coloneqq \sum_{r\geq 0} \frac{ \completeH^*_r(\xvec) }{ \fallingFactorial{u}{r} } = \prod_{i=1}^\infty \frac{u+i}{u+i-x_i}. \]

Define the automorphism $\phi$ on the algebra of symmetric functions as $\phi H^*(u) = H^*(u-1).$

In [OO96] it is then proved that

\[ \schurShifted_{\mu}(\xvec) = \det[ \phi^{j-1} \completeH^*_{\mu_i - i +j} ]_{1\leq i,j \leq l} \qquad \schurShifted_{\mu}(\xvec) = \det[ \phi^{1-j} \elementaryE^*_{\mu'_i - i +j} ]_{1\leq i,j \leq m} \]

where $l \geq \length(\mu)$ and $m \geq \mu_1.$

Littlewood–Richardson rule

Since the factorial Schur polynomials has a Littlewood–Richardson rule [MS99], one can find one for the shifted Schur polynomials as well. For examples of a combinatorial interpretation, see [Mol09].

Theorem.

The shifted Schur Littlewood–Richardson coefficients are defined via

\[ \schurShifted_{\lambda}\schurShifted_{\mu} = \sum_{\nu} c^{\nu}_{\lambda \mu} \schurShifted_{\nu}. \]

Let $\mu, \nu \subseteq \lambda.$ Then

\begin{equation*} c^{\lambda}_{\mu\nu} = \frac{1}{|\lambda|-|\nu|}\left( \sum_{\nu \to \nu^+} c^{\lambda}_{\mu \nu^+} - \sum_{\lambda^- \to \lambda } c^{\lambda^-}_{\mu \nu} \right) \end{equation*}

where the first sum is taken over all possible ways to add one box to the diagram $\nu,$ and the second sum is over all ways to remove one box from $\lambda.$

This together with the identity $c^{\lambda}_{\mu \lambda} = \schurShifted_\mu(\lambda)$ (which can be computed via the Jacobi–Trudi identity) gives a recursive method to compute the $c^{\lambda}_{\mu\nu}.$ This recursion has an analog for the shifted Jack functions.

Note that if $|\nu| = |\lambda|+|\mu|,$ then $c^{\lambda}_{\mu\nu}$ is the classical Littlewood–Richardson coefficient for the Schur functions.

Factorial Schur polynomials

The factorial Schur functions $\schurFactorial_\lambda(\xvec)$ is a family of non-homogeneous symmetric polynomials, indexed by partitions. They were introduced by Biedenharn and Louck in [BL89]. The top-degree component is the classical Schur polynomial $\schurS_\lambda.$ They form a $\setZ$-basis for the space of symmetric polynomials (in $n$ variables).

The factorial Schur functions can be defined using Gelfand–Tsetlin patterns. We sum over all GT-patterns with $n$ rows and top row $\lambda$ (note that this set is in bijection with semi-standard Young tableaux of shape $\lambda$ and maximal entry at most $n$):

\[ \schurFactorial_\lambda(x_1,\dotsc,x_n) \coloneqq \sum_{(m_{ij}) \in GT(\lambda)} \prod_{j=1}^n \prod_{i=1}^j \left( x_j - m_{i,j-1} + i-j \right)_{m_{i,j}-m_{i,j-1}} \]

where we are using the falling factorial notation $(a)_k = a(a-1)\dotsm (a-k+1).$ Under the GT-pattern to SSYT bijection, the quantity $m_{i,j}-m_{i,j-1}$ is sent to the number of times $j$ appears in row $i$ (in the SSYT).

From this definition, we can see that

\[ \schurFactorial_\lambda(x_1,\dotsc,x_n) = \sum_{\mu \downarrow \lambda} \prod_{i=1}^n (x_n - \mu_i - n +i )_{\lambda_i - \mu_i} \cdot \schurFactorial_\mu(x_1,\dotsc,x_{n-1}), \]

where we write $\mu \downarrow \lambda$ for when

\[ \lambda_1 \geq \mu_1 \geq \lambda_2 \geq \mu_2 \geq \dotsb \geq \mu_{n-1} \geq \lambda_n. \]

This is the same as $\mu$ can be in a row immediately below $\lambda$ in a GT-pattern.

The factorial Schur polynomials are special cases of the double Schubert polynomials, indexed by Grassmann permutations.

Skew factorial Schur functions can similarly be defined using sums over skew GT-patterns instead (see [CL93]):

\[ \schurFactorial_{\lambda/\mu}(x_1,\dotsc,x_n) \coloneqq \sum_{(m_{ij}) \in GT(\lambda/\mu)} \prod_{j=1}^n \prod_{i=1}^j \left( x_j - m_{i,j-1} + i-j \right)_{m_{i,j}-m_{i,j-1}}. \]

Here, we sum over GT-patterns with $n+1$ rows, with top row $\lambda$ and bottom row $\mu.$

Alternant quotient formula

We have that

\[ \schurFactorial_{\lambda}(x_1,\dotsc,x_n) = \frac{1}{\Delta(\xvec)} \sum_{\mu} K_{\lambda \mu} \Delta(\xvec-\mu) \prod_{i=1}^n (x_i)_{\mu_i} \]

where $\Delta(\xvec)$ is the Vandermonde determinant, see [Thm. 5, BL90].

Macdonald proved that the above formula leads to the simpler

\[ \schurFactorial_{\lambda}(x_1,\dotsc,x_n) = \frac{\left| (x_i)_{\lambda_j+n-i} \right|}{ \prod_{i \lt j} x_j - x_j \right| }, \]

which is analogous to the alternant quotient formula for Schur functions, see [Thm. 3.2, CL93] for a proof.

Jacobi–Trudi

There is a Jacobi–Trudi identity for skew factorial Schur functions, see [CL93].

Generalized factorial Schur functions

Molev–Sagan (see [MS99]) considers the generalized factorial Schur functions defined via the alternant formula

\[ \schurFactorial_{\lambda}(x_1,\dotsc,x_n | \avec) = \frac{\left| (x_i|a)_{\lambda_j+n-i} \right|}{ \prod_{i \lt j} x_j - x_j \right| }, \]

where $(y|a)_k \coloneqq (y-a_1)(y-a_2)\dotsm (y-a_k).$ Thus, we have that the classical factorial Schur functions are a specialization of the Molev–Sagan family:

\[ \schurFactorial_{\lambda}(x_1,\dotsc,x_n ) = \schurFactorial_{\lambda}(x_1,\dotsc,x_n | 0,1,2,3,\dotsc). \]

These also admits a tableau definition:

\[ \schurFactorial_{\lambda}(x_1,\dotsc,x_n | \avec) = \sum_{T \in \SSYT(\lambda)} \prod_{\alpha \in \lambda} \left( x_{T(\alpha)} - a_{T(\alpha)+c(\alpha)} \right), \]

where $c(\alpha)$ is the content of the box $\alpha.$

These are more or less the same as the double Schur functions, see https://arxiv.org/pdf/0807.2127.pdf, in particular p. 9.

Littlewood–Richardson rule

The factorial Schur functions admits a Littlewood–Richardson rule, see [MS99].

References