MatrixSchubert : Table of Contents

• MatrixSchubert -- matrix Schubert varieties and ASM varieties
  • Investigating matrix Schubert varieties -- basic functions for Schubert determinantal ideals
  • Investigating ASM varieties -- basic functions for alternating sign matrix ideals
    • padASM -- pad an ASM with an identity matrix
    • isPartialASM -- whether a matrix is a partial alternating sign matrix
    • partialASMToASM -- extend a partial alternating sign matrix to an alternating sign matrix
    • rankTable -- compute a table of rank conditions that determines the corresponding ASM or matrix Schubert variety
    • rotheDiagram -- find the Rothe diagram of a partial alternating sign matrix
    • augmentedRotheDiagram -- find the Rothe diagram and rank table for a partial ASM or permutation
    • essentialSet -- compute the essential set in the Rothe Diagram for a partial ASM or a permutation.
    • augmentedEssentialSet -- find the essential set and rank conditions for a partial ASM or a permutation
    • schubertDeterminantalIdeal -- compute an alternating sign matrix ideal (for example, a Schubert determinantal ideal)
    • fultonGens -- compute the Fulton generators of an ASM ideal (for example, a Schubert determinantal ideal)
    • entrywiseMinRankTable -- compute the entrywise minimum rank table of a list of ASMs
    • entrywiseMaxRankTable -- compute the entrywise maximum rank table of a list of ASMs
    • schubertDecompose -- finds the decomposition of an ASM ideal into Schubert determinantal ideals
    • permSetOfASM -- finds the permutation set of an alternating sign matrix
    • isASMIdeal -- whether an ideal is an ASM ideal
    • isASM -- whether a matrix is an ASM
    • isASMUnion -- whether the union of matrix Schubert varieties is an ASM variety
    • getASM -- get the ASM of an ideal (if it exists)
    • isMinRankTable -- whether a matrix is the canonical rank table of some partial ASM
    • rankTableToASM -- to find the a partial ASM associated to a given rank table
    • rankTableFromMatrix -- returns the minimal rank table from an arbitrary integer matrix
    • schubertIntersect -- compute the intersection of ASM ideals
    • schubertAdd -- compute the sum of ASM ideals
    • ASMToMonotoneTriangle -- converts an ASM to a monotone triangle
    • monotoneTriangleToASM -- converts a monotone triangle to an ASM
    • ASMFullList -- lists all ASMs of a fixed size
    • ASMRandomList -- lists random ASMs of a fixed size
    • cohenMacaulayASMsList -- lists all Cohen-Macaulay ASMs of a fixed size which are not permutation matrices
    • nonCohenMacaulayASMsList -- lists all non-Cohen-Macaulay ASMs of a fixed size
    • isIntersectionOfSchubertDeterminantalIdeals -- whether an ideal is the intersection of Schubert determinantal ideals
    • isSchubertCM -- whether an ASM variety is Cohen-Macaulay
    • KPolynomialASM -- compute the K Polynomial of an ASM variety
    • schubertCodim -- compute the codimension (i.e., height) of a Schubert determinantal ideal or ASM ideal
    • schubertRegularity -- compute the Castelnuovo-Mumford regularity of the quotient by an ASM ideal
  • Initial ideals of ASM ideals -- basic functions for investigating initial ideals of ASM varieties
    • antiDiagInit -- compute the (unique) antidiagonal initial ideal of an ASM ideal
    • diagLexInitSE -- Diagonal initial ideal of an ASM ideal with respect to lex, starting from SE corner
    • diagLexInitNW -- Diagonal initial ideal of an ASM ideal with respect to lex, starting from NW corner
    • diagRevLexInit -- Diagonal initial ideal of an ASM ideal with respect to revlex, ordering variables from NW corner
    • subwordComplex -- to find the subword complex associated to w
    • initialIdealsList -- lists all antidiagonal initial ideals of ASMs of a fixed size
  • Functions for investigating permutations -- basic functions for permutations
    • isPerm -- whether a list is a permutation in 1-line notation
    • permToMatrix -- converts a permutation in 1-line notation into a permutation matrix
    • lastDescent -- finds the location of the last descent of a permutation
    • firstDescent -- finds the location of the first descent of a permutation
    • descentSet -- the descent set of a permutation
    • permLength -- to find the length of a permutation in 1-line notation.
    • inverseOf -- to return the inverse of a permutation in 1-line notation.
    • longestPerm -- to return the longest permutation of length n
    • toOneLineNotation -- rewrites a transposition or permutation matrix in 1-line notation
    • composePerms -- computes the composition of two permutations
    • isPatternAvoiding -- whether a permutation avoids certain patterns, e.g. $2143$-avoiding or $312$- and $231$-avoiding
    • isVexillary -- whether a permutation is vexillary, i.e. 2143-avoiding
    • avoidsAllPatterns -- whether a permutation avoids all of the given patterns
    • isCartwrightSturmfels -- whether a permutation is Cartwright-Sturmfels
    • isCDG -- whether a permutation is CDG
    • rajcode -- finds the Rajchgot code of a permutation
    • rajIndex -- finds the Rajchgot index of a permutation
    • grothendieckPolynomial -- computes the Grothendieck polynomial of a permutation
    • schubertPolynomial -- computes the Schubert polynomial of a permutation
    • doubleSchubertPolynomial -- computes the double Schubert polynomial of a permutation
    • PipeDream -- the class representing a pipe dream
      • pipeDreams -- computes the set of reduced pipe dreams corresponding to a permutation
      • pipeDreamsNonReduced -- computes the set of all pipe dreams corresponding to a permutation