SageMath code for computing the dimensions of the weight spaces of the cyclotomic KLR algebras attached to symmetrisable quivers using the amazing formula
from the paper Graded dimensions and monomial bases for the cyclotomic quiver Hecke algebras of Hu and Shi. Here,
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 orC
can be a SageMath Cartan type, such asCartanType(['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 ann
-tuple of vertices of the quiver -
j
is ann
-tuple of vertices of the quiver. Ifj
is omitted thenj
is set equal toi
. -
verbose
an optional parameter that when set toTrue
causes additional information from the Hu-Shi formula to be printed.
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
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 theklr_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)
Andrew Mathas Copyright (C) 2022
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.