GradedDimKLR

SageMath code for computing the dimensions of the weight spaces of the cyclotomic KLR algebras attached to symmetrisable quivers using the amazing formula

$$\dim_q e(i)R^\Lambda_n e(j) = \sum_{w\in S_{i,j}} \prod_{t=1}^n[N^\Lambda(w,i,t)]_{i_t} q_{i_t}^{N^\Lambda(i,t)-1}$$

from the paper Graded dimensions and monomial bases for the cyclotomic quiver Hecke algebras of Hu and Shi. Here, $e(i)R^\Lambda_ne(j)$ is the weight space of the KLR algebra $R^\Lambda_n$ determined by the sequences $i,j\in I^n$ and $\Lambda$ is a dominant weight for the corresponding quiver, which has vertex set $I$. See the Hu-Shi paper for the unexplained notation in the formula.

The attached code provides a single command klr_cyclotomic_dimension that uses the Hu-Shi formula to compute the dimension. The syntax of this command is:

   klr_cyclotomic_dimension(C, L, i, [j, verbose])

where:

• C specifies the Cartan type of the quiver. This can either be a list of the form ['A', 3] (finite type $A_3$), ['A', 3, 1] (affine type $A_3^{(1)}$), ['B',4] (finite type $B_4$), etc or C can be a SageMath Cartan type, such as CartanType(['D',3,1]).

• L is a list that specifies the dominant weight. For example, [0,2,2,3] represents the dominant weight $\Lambda_0+2\Lambda_2+\Lambda_3$.

• i is an n-tuple of vertices of the quiver

• j is an n-tuple of vertices of the quiver. If j is omitted then j is set equal to i.

• verbose an optional parameter that when set to True causes additional information from the Hu-Shi formula to be printed.

Examples

    sage: klr_cyclotomic_dimension(['D',4],[2], [2,3,4,1], verbose=True)
    Subgroup of permutations = < () >
    N(1,t)-1: 0  0  0  0
    N(w,t):   1  1  1  1
    X(w): 1
    1
    sage: klr_cyclotomic_dimension(['D',4],[2], [2,3,4,1], [2,3,4,1], verbose=True)
    Subgroup of permutations = < () >
    N(1,t)-1: 0  0  0  0
    N(w,t):   1  1  1  1
    X(w): 1
    1
    sage: klr_cyclotomic_dimension(['D',4],[2], [2,3,4,1], [2,4,3,1], verbose=True)
    Subgroup of permutations = < (1,2) >
    N(1,t)-1: 0  0  0  0
    N(w,t):   1  1  1  1
    X(w): 1
    1
    sage: klr_cyclotomic_dimension(['A',2,1],[0], [0,1,2], verbose=True)
    Subgroup of permutations = < () >
    N(1,t)-1: 0  0  1
    N(w,t):   1  1  2
    X(w): q^2 + 1
    q^2 + 1
    sage: klr_cyclotomic_dimension(['A',2,1],[0], [0,2,1], verbose=True)
    Subgroup of permutations = < () >
    N(1,t)-1: 0  0  1
    N(w,t):   1  1  2
    X(w): q^2 + 1
    q^2 + 1
    sage:
    sage: klr_cyclotomic_dimension(['A',2,1],[0], [0,1,2], [0,2,1], verbose=True)
    Subgroup of permutations = < (1,2) >
    N(1,t)-1: 0  0  1
    N(w,t):   1  1  1
    X(w): q
    q
    sage: klr_cyclotomic_dimension(['A',1],[1,1],[1],[1], verbose=True)
    Subgroup of permutations = < () >
    N(1,t)-1: 1
    N(w,t):   2
    X(w): q^2 + 1
    q^2 + 1
    sage: klr_cyclotomic_dimension(['A',1],[1,1],[1],[1], verbose=True)
    Subgroup of permutations = < () >
    N(1,t)-1: 1
    N(w,t):   2
    X(w): q^2 + 1
    q^2 + 1
    sage: klr_cyclotomic_dimension(['B',3],[2], [2,3,3,2,1], verbose=True)
    Subgroup of permutations = < (), (1,2), (0,3), (0,3)(1,2) >
    N(1,t)-1: 0  1 -1  0  1
    N(w,t):   1  2  0  1  2
    X(w): 0
    N(w,t):   1  2  2  1  2
    X(w): (q^4 + 1)*(q^2 + 1)^2/q^2
    N(w,t):   1  0 -2  1  2
    X(w): 0
    N(w,t):   1  0  0  1  2
    X(w): 0
    (q^4 + 1)*(q^2 + 1)^2/q^2

Usage

• With a local installation of SageMath, start SageMath and attach the file graded_dim_klr using:

   sage: %attach graded_dim_klr
   sage: klr_cyclotomic_dimension(['A',3],[2], [2,3,3,2,1], [2,3,2,3,1])
   sage: klr_cyclotomic_dimension(['B',3],[2], [2,3,3,2,1], verbose=True)

• Using SageMathCell, cut-and-paste the code from graded_dim_klr into a cell and then type the klr_cyclotomic_dimension commands into the bottom of the cell.

   klr_cyclotomic_dimension(['A',3],[2], [2,3,3,2,1], [2,3,2,3,1])
   klr_cyclotomic_dimension(['B',3],[2], [2,3,3,2,1], verbose=True)

Author

Andrew Mathas Copyright (C) 2022

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GNU General Public License, Version 3, 29 June 2007

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License GPL as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.