Math.Combinatorics.Kostka

This module provides some functions to compute Kostka-Jack numbers, i.e. Kostka numbers with a Jack parameter, possibly skew.

Kostka numbers

kostkaNumbersWithGivenLambda

Arguments

:: Partition

the integer partition lambda

-> Rational

Jack parameter

-> Map Partition Rational

Kostka numbers \(K_{\lambda,\mu}(\alpha)\) with Jack parameter, or Kostka-Jack numbers, for a given integer partition \(\lambda\) and a given Jack parameter \(\alpha\). These are the ordinary Kostka numbers when \(\alpha=1\). The function returns a map whose keys represent the partitions \(\mu\) and the value attached to a partition \(\mu\) is the Kostka-Jack number \(K_{\lambda,\mu}(\alpha)\). The partition \(\mu\) is included in the keys of this map if and only if \(K_{\lambda,\mu}(\alpha) \neq 0\). The Kostka-Jack number \(K_{\lambda,\mu}(\alpha)\) is the coefficient of the monomial symmetric polynomial \(m_\mu\) in the expression of the \(P\)-Jack polynomial \(P_\lambda(\alpha)\) as a linear combination of monomial symmetric polynomials.

kostkaNumbers

Arguments

:: Int

weight of the partitions

-> Rational

Jack parameter

-> Map Partition (Map Partition Rational)

Kostka numbers \(K_{\lambda,\mu}(\alpha)\) with Jack parameter, or Kostka-Jack numbers, for a given weight of the partitions \(\lambda\) and \(\mu\) and a given Jack parameter \(\alpha\). These are the ordinary Kostka numbers when \(\alpha=1\). The function returns a map whose keys represent the partitions \(\lambda\) and the value attached to a partition \(\lambda\) represents the map \(\mu \mapsto K_{\lambda,\mu}(\alpha)\) where the partition \(\mu\) is included in the keys of this map if and only if \(K_{\lambda,\mu}(\alpha) \neq 0\). The Kostka-Jack number \(K_{\lambda,\mu}(\alpha)\) is the coefficient of the monomial symmetric polynomial \(m_\mu\) in the expression of the \(P\)-Jack polynomial \(P_\lambda(\alpha)\) as a linear combination of monomial symmetric polynomials.

symbolicKostkaNumbersWithGivenLambda

Arguments

:: Partition

the integer partition lambda

-> Map Partition RatioOfQSprays

Kostka-Jack numbers \(K_{\lambda,\mu}(\alpha)\) with symbolic Jack parameter \(\alpha\) for a given integer partition \(\lambda\). This function returns a map whose keys represent the partitions \(\mu\) and the value attached to a partition \(\mu\) is the Kostka-Jack number \(K_{\lambda,\mu}(\alpha)\). The partition \(\mu\) is included in the keys of this map if and only if \(K_{\lambda,\mu}(\alpha) \neq 0\).

symbolicKostkaNumbers

Arguments

:: Int

weight of the partitions

-> Map Partition (Map Partition RatioOfQSprays)

Kostka-Jack numbers \(K_{\lambda,\mu}(\alpha)\) with symbolic Jack parameter \(\alpha\) for a given weight of the partitions \(\lambda\) and \(\mu\). This function returns a map whose keys represent the partitions \(\lambda\) and the value attached to a partition \(\lambda\) represents the map \(\mu \mapsto K_{\lambda,\mu}(\alpha)\) where the partition \(\mu\) is included in the keys of this map if and only if \(K_{\lambda,\mu}(\alpha) \neq 0\).

skewKostkaNumbers

Arguments

:: Rational

Jack parameter

-> Partition

outer partition of the skew partition

-> Partition

inner partition of the skew partition

-> Map Partition Rational

Skew Kostka numbers \(K_{\lambda/\mu, \nu}(\alpha)\) with a given Jack parameter \(\alpha\) and a given skew partition \(\lambda/\mu\). For \(\alpha=1\) these are the ordinary skew Kostka numbers. The function returns a map whose keys represent the partitions \(\nu\). The skew Kostka-Jack number \(K_{\lambda/\mu, \nu}(\alpha)\) is the coefficient of the monomial symmetric polynomial \(m_\nu\) in the expression of the skew \(P\)-Jack polynomial \(P_{\lambda/\mu}(\alpha)\) as a linear combination of monomial symmetric polynomials. Note: the skew Kostka-Jack numbers \(K_{\lambda/\mu, \nu}(\alpha)\) are well defined when the Jack parameter \(\alpha\) is zero, however this function does not work for \(\alpha=0\); a possible way to get the skew Kostka-Jack numbers \(K_{\lambda/\mu, \nu}(0)\) is to use the function symbolicSkewKostkaNumbers to get the skew Kostka-Jack numbers with a symbolic Jack parameter \(\alpha\), and then to substitute \(\alpha\) with \(0\).

symbolicSkewKostkaNumbers

Arguments

:: Partition

outer partition of the skew partition

-> Partition

inner partition of the skew partition

-> Map Partition RatioOfQSprays

Skew Kostka numbers \(K_{\lambda/\mu, \nu}(\alpha)\) with symbolic Jack parameter \(\alpha\) for a given skew partition \(\lambda/\mu\). This function returns a map whose keys represent the partitions \(\nu\).