Documentation

Permutations

The grobnercrystals.Perm class is a class for permutations which includes useful methods for working with matrix Schubert varieties:

class grobnercrystals.Perm(perm, region=None)

A class for permutations

Parameters:

  • perm (list) – The permutation in one-line notation as a list of integers

  • region (list, optional) – The region the permutation should be restricted to, as a list of tuples (i,j) with i,j in [0,length(perm)-1] (for working with usual instead of effective matrix Schubert varieties)

Variables:

  • perm – the permutation in one-line notation as a list of integers

  • l – int, length of the permutation

  • region – parameter region if specified, else list of tuples (i,j) of integers between 0 and l-1

  • descents – list of the descents of the permutation (as integers)

  • inverse – inverse of the permutation in one-line notation as a list of integers

  • inverse_descents – list of the descents of the inverse of the permutation (as integers)

  • vexillary – bool, True if the permutation is vexillary, else False

  • max_row – int, largest i such that (i,j) is in region

  • max_col – int, largest j such that (i,j) is in region

eff_reg(retMaxDim=False)

Calculates the effective region of the permutation, i.e. the set of all tuples (i,j) so that (i,j) is weakly northwest of the southeast most box in the Rothe diagram for the permutation.

Parameters:

retMaxDim (bool, optional) – whether to return the largest row and column in the effective region, defaults to False

ess_set()

Finds Fulton’s essential set.

Returns:

Fulton’s essential set as a list of indices [(i,j)]

Return type:

list

filled_ess_set()

Finds Fulton’s essential set, together with the rank conditions imposed by each box in the essential set.

Returns:

A list [[essential set box, number in that box]]

Return type:

list

levi_datum()

Finds the Levi datum corresponding to the largest Levi group which acts on the effective matrix Schubert variety corresponding to the permutation. If a region is specified, returns the Levi datum corresponding to the largest Levi group which acts on the matrix Schubert variety when restricted to that region.

Returns:

A Levi datum [I,J], where I,J are lists of integers

Return type:

list

Filtered RSK

Grobnercrystals includes a function for computing filteredRSK of a matrix with respect to a given Levi datum.

Note

This function takes numpy matrices as input.

grobnercrystals.filtered_RSK()

Returns the result of applying filteredRSK to a matrix for a given Levi datum.

Parameters:

  • M (numpy matrix) – The matrix to be filterRSK’d

  • I (list) – The row Levi datum

  • J (list) – The column Levi datum

Returns:

A list of tableau [[row tableaux],[column tableaux]]

Return type:

list

Crystal Operators on Matrices

Note

These functions take numpy matrices as input.

grobnercrystals.e()

Raising matrix crystal operator.

Parameters:

  • M (numpy matrix) – Matrix to apply the operator to

  • i (int) – row or column index

  • rc (str or int) – ‘r’ or 0 if row operator, ‘c’ or 1 if column operator

Returns:

Result of applying raising crystal operator to M if possible. If the result of applying the operator to M is the empty symbol, return None.

Return type:

numpy matrix or None

grobnercrystals.f()

Lowering matrix crystal operator.

Parameters:

  • M (numpy matrix) – Matrix to apply the operator to

  • i (int) – row or column index

  • rc (str or int) – ‘r’ or 0 if row operator, ‘c’ or 1 if column operator

Returns:

Result of applying lowering crystal operator to M if possible. If the result of applying the operator to M is the empty symbol, return None.

Return type:

numpy matrix or None

Polynomial Rings

Grobnercrystals includes a class for defining a polynomial ring in an m by n matrix of variables.

class grobnercrystals.PolRing(m, n, name='z', to='GRevLex')

Class for polynomial rings in an m by n matrix of variables.

Parameters:

  • m (int) – number of rows in the matrix of variables

  • n (int) – number of columns in the matrix of variables

  • name (str) – variable names, optional, default ‘z’

  • to (str) – Macaulay2 ring term order, optional, default ‘GRevLex’

Variables:

  • R – Sage PolynomialRing in an m by n matrix of variables

  • Z – matrix of variables

  • xR – Sage PolynomialRing of m ‘x’ variables x1,…,xm

  • yR – Sage PolynomialRing of n ‘y’ variables y1,…,yn

  • X – generators of xR, output of xR.gens()

  • Y – generators of yR, output of yR.gens()

  • graded_m2_str – str containing Macaulay2 command for generating R (with multigrading)

  • ungraded_m2_str – str containing Macaulay2 command for generating R (no multigrading)

hilb_exp(deg)

Returns Hilbert series for R up to specified degree as a SplitPoly object.

Parameters:

deg (int) – degree of the expansion

Returns:

Hilbert series of R up to degree deg

Return type:

SplitPoly

Ideals and Bicrystalline Tests

Ideals are constructed using the grobnercrystals.BIdeal class. Every BIdeal is an ideal of a polynomial ring in an m by n matrix of variables, implemented as a PolRing object. This class has a .bicrystalline() method for checking whether the ideal is bicrystalline

Note

The .bicrystalline() method does not check whether the ideal is stable under the given Levi group, only whether the set of its non-standard monomials is closed under all admissible bicrystal operators for that Levi group. See Group Action Checks.

class grobnercrystals.BIdeal(gens, PR)

Class for ideals of PolRing objects.

Parameters:

  • gens (polynomials in PR.R) – generators of the ideal

  • PR (PolRing) – ring in which the ideal lives

Variables:

  • I – ideal of PR.R generated by gens

  • R – alias of PR.R

  • X – alias of PR.X

  • Y – alias of PR.Y

  • m2_ideal_str – str containing command for defining I in Macaulay2

bicrystalline(I, J, use_to=None, detailed_output=False)

Checks whether the ideal is bicrystalline with respect to a given Levi action. Optionally, instead of checking all term orders, one can check a specific term order. This function does not check whether the ideal is stable under the action of the given Leiv group, or whether it is homogeneous.

Parameters:

  • I (list) – row Levi datum

  • J (list) – column Levi datum

  • use_to (list, optional) – Macaulay2 term order to use, defaults to None

  • detailed_output (bool, optional) – when true, outputs details about the checks it makes to the terminal, defaults to False

Returns:

True if the ideal is bicrystalline (assuming it is homogeneous and stable under the input Levi group), else False

Return type:

bool

def_m2_vars(graded=False, ideal_name='I', ring_name='R')

Define the ideal in Macaulay2. After running this command, one can run commands referencing the ideal “I” in Macaulay2.

Parameters:

  • graded (bool, optional) – whether the Macaulay2 ideal should be graded, defaults to False

  • ideal_name (str, optional) – what the ideal should be called in Macaulay2, defaults to “I”

  • ring_name (str, optional) – what the ring should be called in Macaulay2, defaults to “R”

gb(to=None)

Computes a Grobner basis for the ideal. Optionally, one may specify the term order. The default is the term order of the parent PolRing.

Parameters:

to (str, optional) – Macaulay2 term order, defaults to None

Returns:

list of Grobner basis generators as ring elements

Return type:

list

gb_lts(to=None)

Computes the lead terms of a Grobner basis for the ideal. Optionally, one may specify the term order. The default is the term order of the parent PolRing.

Parameters:

to (str, optional) – Macaulay2 term order, defaults to None

Returns:

list of lead terms of Grobner basis generators as ring elements

Return type:

list

gb_lts_mats(to=None)

Computes the lead terms of a Grobner basis for the ideal. Optionally, one may specify the term order. The default is the term order of the parent PolRing. The lead terms are output as numpy matrices.

Parameters:

to (str, optional) – Macaulay2 term order, defaults to None

Returns:

list of lead terms of Grobner basis generators as numpy matrices

Return type:

list

hilb_exp(deg)

Computes the Hilbert series expansion of I up to a given degree. The Hilbert series is output as a SplitPol.

Parameters:

deg (int) – degree

Returns:

Hilbert series up to a given degree

Return type:

SplitPol

min_test_set(op, gb=None, to=None)

Outputs the unique minimal test set for a given bicrystal operator, and optionally a given Grobner basis or term order.

Parameters:

  • op (list) – bicrystal operator as a list [‘f’ or ‘e’,i,’r’ or ‘c’], e.g. [‘f’,1,’r’] for the row lowering operator f1

  • gb (list, optional) – lead terms of a Grobner basis for I as a list of numpy matrices, defaults to None

  • to (str, optional) – a Macaulay2 term order, defaults to None

Returns:

the minimal test set for I as a list of numpy matrices

Return type:

list

nstd_mons(deg, gens=None, leq=False)

Returns all non-standard monomials for I up to a given degree as ring elements.

Parameters:

  • deg (int) – degree

  • gens (list, optional) – optional list of lead terms of generators of the initial ideal as ring elements, defaults to None

  • leq (bool, optional) – optionally return all nonstandard monomials of at most input degree, defaults to False

Returns:

list of all nonstandard monomials of a given degree as ring elements

Return type:

list

nstd_mons_mats(deg, gens=None, leq=False)

Returns all non-standard monomials for I up to a given degree as numpy matrices.

Parameters:

  • deg (int) – degree

  • gens (list, optional) – optional list of lead terms of generators of the initial ideal as ring elements, defaults to None

  • leq (bool, optional) – optionally return all nonstandard monomials of at most input degree, defaults to False

Returns:

list of all nonstandard monomials of a given degree as numpy matrices

Return type:

list

test_set(op, gb=None, to=None)

Outputs a test set for a given bicrystal operator, and optionally a given Grobner basis or term order.

Parameters:

  • op (list) – bicrystal operator as a list [‘f’ or ‘e’,i,’r’ or ‘c’], e.g. [‘f’,1,’r’] for the row lowering operator f1

  • gb (list, optional) – lead terms of a Grobner basis for I as a list of numpy matrices, defaults to None

  • to (str, optional) – a Macaulay2 term order, defaults to None

Returns:

a test set for I as a list of numpy matrices

Return type:

list

Interesting Classes of Ideals

Grobnercrystals provides functions for defining several ideals of interest.

grobnercrystals.shape_ideal()

Outputs the BIdeal I generated by all bitableaux of shape l in the polynomial ring in a matrix of m by n variables.

Parameters:

  • l (list) – a partition, as a decreasing list of integers

  • m (int) – number of rows of the matrix of variables

  • n (int) – number of columns of the matrix of variables

  • R (PolRing, optional) – PolRing in which to define the ideal, defaults to None

Returns:

shape ideal I

Return type:

BIdeal

grobnercrystals.classical_det_ideal()

Creates the BIdeal I of k by k minors of an m by n matrix.

Parameters:

  • m (int) – number of rows in the matrix of variables

  • n (int) – number of columns in the matrix of variables

  • k (int) – size of minors

Returns:

classical determinantal ideal

Return type:

BIdeal

grobnercrystals.msv()

Creates the BIdeal I defining the matrix Schubert variety corresponding to a permutation w.

Parameters:

  • w (list) – permutation in one-line notation as a list of integers

  • region (list, optional) – list of tuples (i,j) defining a region in which to define the ideal, defaults to None

  • R (PolRing, optional) – the ring in which I is to be defined, defaults to None

Returns:

ideal defining the matrix Schubert variety for w

Return type:

BIdeal

grobnercrystals.eff_msv()

Creates the effective matrix Schubert ideal corresponding to w. The effective matrix Schubert ideal lives in the ring of r by c matrices, where r is the largest row in which the Rothe diagram for w has a box and c is the largest column in which the Rothe diagram for w has a box. The ideal is generated by the Fulton generators, together with all variables outside the effective region (i.e. all variables zij for which (i,j) is strictly southeast of any box in the Rothe diagram for w).

Parameters:

w (list) – permutation in one-line notation as a list of integers

Returns:

ideal defining the effective matrix Schubert variety for w

Return type:

BIdeal

grobnercrystals.mrv()

Constructs the matrix Richardson ideal corresponding to input permutations u and v.

Parameters:

  • u (list) – a permutation in one-line notation as a list of integers

  • v (list) – a permutation in one-line notation as a list of integers

Returns:

the matrix Richardson ideal corresponding to u and v

Return type:

BIdeal

The following function is useful for defining various determinantal ideals.

grobnercrystals.minors()

Outputs the k by k minors of the matrix M of size k. Optionally, outputs the k by k minors of a submatrix B of M, input as [[list of rows of B],[list of columns of B]].

Parameters:

  • M (Sage matrix) – matrix

  • k (int) – minor size

  • B (list, optional) – submatrix of M, input as [[list of rows of B],[list of columns of B]] (row and column numbers are 0-indexed), defaults to None

Returns:

list of minors

Return type:

list

Group Action Checks

grobnercrystals.check_action()

Checks whether a given Levi group acts on a given ideal.

Parameters:

  • I (BIdeal) – The ideal to be checked

  • L (list) – A Levi datum [I,J]

  • detailed_output (bool, optional) – If True, prints a pair (Gröbner basis element,Gröbner basis element after group action) which fails the check (if such a pair exists), defaults to False

Returns:

True if the Levi group defined by L acts on I, else False

Return type:

bool

Polynomials and Schur Expansions

Grobnercrystals includes a special class SplitPoly for polynomials in two sets of variables. This class has a .expand() method for computing split-Schur expansions of such polynomials.

class grobnercrystals.SplitPoly(poly, xR, yR)

A class for polynomials in two sets of variables (‘x’ variables and ‘y’ variables). The rings in which the ‘x’ and ‘y’ variables live must be specified.

Parameters:

  • poly (class 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular') – the polynomial as an element of a Sage MPolynomialRing

  • xR (class 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomialRing_libsingular') – the ring of ‘x’ variables

  • yR (class 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomialRing_libsingular') – the ring of ‘y’ variables

Variables:

  • X – the generators of xR (output of xR.gens())

  • Y – the generators of yR (output of yR.gens())

  • x – list of strings [‘xi’] of ‘x’ variables

  • y – list of strings [‘yi’] of ‘y’ variables

  • deg – degree of the polynomial

  • d – dictionary for the polynomial of the form {degree : homogeneous component of the polynomial with that degree}

expand(x=None, y=None, I=[], J=[], returnDict=False, maxIter=10000)

Expands the polynomial into a sum of products of Schur polynomials based on a given Levi datum. If no inputs are given, returns the polynomial. If the user specifies x=’m’ or y=’m’, the corresponding variables will be left as monomials. The default output is a string representing the expansion. Optionally, can output a dictionary of the form {tuple of partitions, permutations, or exponents : coefficient in the expansion}.

Parameters:

  • x (str, optional) – What type of polynomial to expand the x variables in (‘s’ for Schur,’m’ for Monomial), defaults to None

  • y (str, optional) – What type of polynomial to expand the x variables in (‘s’ for Schur,’m’ for Monomial), defaults to None,

  • I (list, optional) – Row Levi datum, defaults to []

  • J (list, optional) – Column Levi datum, defaults to []

  • returnDict (bool, optional) – True to return a dictionary, False to return a string, defaults to False

  • maxIter (int, optional) – Maximum number of subtractions allowed per degree component, defaults to 10000

Raises:

MemoryError – if the number of subtractions exceeds maxIter

Returns:

A string representing the desired expansion (or, optionally, a dictionary)

Return type:

str (optionally, dict)