The symmetric functions catalog

An overview of symmetric functions and related topics

2020-08-24

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.

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.

References