Math.Algebra.Combinatorics

This module provides some functions to compute Kostka numbers with a Jack parameter, possibly skew, some functions to enumerate semistandard tableaux, possibly skew, with a given shape and a given weight, and a function to enumerate the Gelfand-Tsetlin patterns defined by a skew partition.

Kostka numbers

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 standard Kostka numbers when \(\alpha=1\)). This 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.

symbolicKostkaNumbers

Arguments

:: Int

weight of the partitions

-> Map Partition (Map Partition RatioOfQSprays)

Kostka numbers \(K_{\lambda,\mu}(\alpha)\) with symbolic Jack parameter \(\alpha\) for a given weight of the partitions \(\lambda\) and \(\mu\). This 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.

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 returns a map whose keys represent the partitions \(\nu\).

Tableaux

semiStandardTableauxWithGivenShapeAndWeight

Arguments

:: Partition

shape, integer partition

-> [Int]

weight

-> [[[Int]]]

Semistandard tableaux with a given shape (an integer partition) and a given weight vector. The weight is the vector whose i-th element is the number of occurrences of i in the tableau.

skewTableauxWithGivenShapeAndWeight

Arguments

:: Partition

outer partition of the skew partition

-> Partition

inner partition of the skew partition

-> [Int]

weight

-> [SkewTableau Int]

Skew semistandard tableaux with a given shape (a skew partition) and a given weight vector. The weight is the vector whose i-th element is the number of occurrences of i in the tableau.

Gelfand-Tsetlin patterns

skewGelfandTsetlinPatterns

Arguments

:: Partition

outer partition of the skew partition

-> Partition

inner partition of the skew partition

-> [Int]

weight

-> [[Partition]]

Skew Gelfand-Tsetlin patterns defined by a skew partition and a weight vector.