GTPatterns
Gelfand–Tsetlin patterns (skew, row-flagged, cylindric), BZ patterns, Gog and Magog patterns, tiles and snakes, lattice paths, stretched Kostka (Ehrhart) data, TikZ output.
Load with Needs["GTPatterns`"].
Background on symmetricfunctions.com: Gelfand–Tsetlin patterns and polytopes, Hives and BZ-polytopes, Ehrhart theory.
Functions and symbols
BoxCountMatrix
BoxCountMatrix[GTPattern[rows]] returns the matrix whose entry in row i and column j counts entries j in tableau row i, with GT rows ordered bottom to top.
Background: Gelfand–Tsetlin patterns and polytopes
BZPattern
BZPattern[data] represents a Berenstein-Zelevinsky pattern.
Background: Hives and BZ-polytopes
BZPatterns
BZPatterns[lam, mu, nu] returns BZ-patterns counted by the Littlewood-Richardson coefficient c^lam_{mu,nu}.
Background: Hives and BZ-polytopes
BZPlus
BZPlus[b1,b2,…] adds BZ-patterns entrywise.
Background: Hives and BZ-polytopes
ContainingFaceDimension
ContainingFaceDimension[g] returns the dimension of the face of the GT polytope containing g.
Background: Gelfand–Tsetlin patterns and polytopes
EnableSkew
EnableSkew is an option for ShapeTriplets, GTTiles, GTPatternForm, and GTPatternTikz controlling whether skew cells are included.
Background: Gelfand–Tsetlin patterns and polytopes
GogPatterns
GogPatterns[n] returns the Gog patterns of size n. GogPatterns[n,k] uses k columns.
Background: Gelfand–Tsetlin patterns and polytopes
GTEhrhartPolynomial
GTEhrhartPolynomial[lam,mu,w,k] counts GT-patterns of (k lam,k mu,k w) when k is an integer; a symbolic k returns the interpolating Ehrhart polynomial.
Background: Ehrhart theory
GTMonomial
GTMonomial[g,x] returns the monomial in x associated with the weight of g.
Background: Gelfand–Tsetlin patterns and polytopes
GTPartition
GTPartition is an option for GTPatternForm and GTPatternTikz. Its values are “Tiles”, “Snakes”, “FreeTiles”, “ShadedTiles”, or None.
Background: Gelfand–Tsetlin patterns and polytopes
GTPattern
GTPattern[data] represents a GT-pattern as a list of rows (partitions) ordered bottom to top.
GTPattern[YoungTableau[t]] converts a (skew) SSYT to its GT-pattern.
YoungTableau[gtp] converts a GT-pattern back to a SSYT.
gtp[r,c] accesses the entry at row r, column c (1-indexed; row 1 = bottom = inner shape).
Background: Gelfand–Tsetlin patterns and polytopes
GTPatternForm
GTPatternForm[gtp] returns the graphical representation of the GT-pattern.
Background: Gelfand–Tsetlin patterns and polytopes
GTPatterns
GTPatterns[lam,mu,w] returns a list of all GT-patterns with outer shape lam, inner shape mu (default {}), and weight vector w (default {}), corresponding to SSYT of skew shape lam/mu with content w.
Optional argument cylindricShift (default Infinity) restricts to cylindric GT-patterns with the given column shift.
Option RowFlags->{{a1,b1},{a2,b2},…} constrains entries in SSYT row r to the range [ar,br] (default {1,Infinity} = no constraint).
Background: Gelfand–Tsetlin patterns and polytopes
GTPatternTikz
GTPatternTikz[g] returns TikZ code for a GT-pattern.
Background: Gelfand–Tsetlin patterns and polytopes
GTPlus
GTPlus[g1,g2,…] adds Gelfand-Tsetlin patterns entrywise, padding smaller patterns.
Background: Gelfand–Tsetlin patterns and polytopes
GTShape
GTShape[gtp] returns {lam, mu, w} for a GT-pattern gtp, where lam is the outer shape, mu is the inner shape (empty list for straight shapes), and w is the weight vector.
Background: Gelfand–Tsetlin patterns and polytopes
GTSnakes
GTSnakes[g] returns the equal-entry snakes of a GT-pattern.
Background: Gelfand–Tsetlin patterns and polytopes
GTTiles
GTTiles[g] returns {freeTiles, fixedTiles} for a GT-pattern.
Background: Gelfand–Tsetlin patterns and polytopes
KostkaRange
KostkaRange is an option for ShapeTriplets specifying the minimum and maximum allowed Kostka multiplicity.
Background: Gelfand–Tsetlin patterns and polytopes
LatticePathForm
LatticePathForm[g] or LatticePathForm[tab] returns graphics for the non-intersecting lattice paths.
Background: Gelfand–Tsetlin patterns and polytopes
LatticePathTikz
LatticePathTikz[g] or LatticePathTikz[tab] returns TikZ code for the non-intersecting lattice paths.
Background: Gelfand–Tsetlin patterns and polytopes
MagogPatterns
MagogPatterns[n] returns the Magog patterns of size n. MagogPatterns[n,k] uses k columns.
Background: Gelfand–Tsetlin patterns and polytopes
RowFlags
RowFlags is an option for GTPatterns that restricts the entries in tableau row r to a specified inclusive interval {ar,br}.
Background: Gelfand–Tsetlin patterns and polytopes
ShapeTriplets
ShapeTriplets[lambda, options] returns {lambda, mu, w} triples for skew shapes lambda/mu and weights w. EnableSkew, WeightRange, and KostkaRange control skew shapes, weight sizes, and Kostka multiplicities.
Background: Gelfand–Tsetlin patterns and polytopes
TilingMatrix
TilingMatrix[g] returns the tiling matrix of a GT-pattern.
Background: Gelfand–Tsetlin patterns and polytopes
UpperBoundKostkaDegree
UpperBoundKostkaDegree[lam,mu,w] returns an upper bound for the degree of the stretched Kostka polynomial.
Background: Ehrhart theory
WeightRange
WeightRange is an option for ShapeTriplets specifying the minimum and maximum number of parts in a weight.
Background: Gelfand–Tsetlin patterns and polytopes