Compute dimensions of cyclotomic (weighted) KLR(W)

This is SageMath code to compute dimensions of cyclotomic KLRW algebras. The code is due to Andrew Mathas (all the credits should go to Andrew!) and the original file and explanation can be found here: click. The purpose of this side is to give another explanation of how the code works and is also a (very mildly) updated version of the code.

Background

The starting point is an amating paper by Hu and Shi click that proves the following dimension formula for cyclotomic KLR algebras:

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

Here $R^\Lambda_n$ denotes the cyclotomic KLR algebra with $n$-strands, and $e(i)$ and $e(j)$ are idempotents corresponding to residue sequences. The sum runs over the subset of the symmteric group $S_n$ of permuations from $i$ to $j$. The $q$ indicates the grading and the $[N^\Lambda(w,i,t)]_{i_t}$ are certain quantum numbers that can be negative (careful with cancellations).

What the code can do - basics

The code that can be downloaded here (klrw-dim.pv) 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, bi, bj=None, base=[], verbose=False)

where:

• The first entry C specifies the Cartan type of the quiver that is used for the KLR algebra. The Crtan types are as in [CartanType]{https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/cartan_type.html} so for example ['A', 3] (finite type $A_3$), ['A', 3, 1] (affine type $A_3^{(1)}$), ['B',4] (finite type $B_4$), etc.

• L is a list that specifies the dominant weight, that is, the number and the labels of the red strings in the KLRW world. 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, the residue sequence = coloring of the strings at the, say, bottom of a diagram. (This does not need to be an n-tuple, see base below. Ditto for the next entry.)

• j is an n-tuple of vertices of the quiver, the residue sequence = coloring of the strings at the, say, top of a diagram. If j is omitted, then j is set equal to i.

• base is a tuple that is similar to i and j and serves as the base. If this is not empty (it is empty by default), then the residue sequences are actually the concatenations of base and i respectively j. The idea is that the Hu-Shi formula runs voer large symmetric groups, which might be outside of the calculation scope. With a nontrivial base the calculation of the permutations is restricted to i and j only which makes the calculation faster. This however should be treated with care as the Hu-Shi formual is not local.

• verbose an optional parameter that when set to True gives additional output that might be useful to analize what is going on.

Before we do examples, let me explain how to get started.

Getting started

• Using the online calculator SageMathCell is easy: Cut-and-paste the text from klrw-dim.py into a cell and then type the klr_cyclotomic_dimension commands into the bottom of the cell.

   klr_cyclotomic_dimension(['A',3],[2,3], [2,3,3,2,1], [2,3,2,3,1])

• 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],[2,3], [2,3,3,2,1], [2,3,2,3,1])

Examples

Ok, let us have a look at

   klr_cyclotomic_dimension(['A',3],[2,3], [2,3,3,2,1], [2,3,2,3,1])

The output is

   ({(q + 1/q)^2*q: [(0,2,1,3)]}, (q^2 + 1)^2/q)

This is saying that there is only one diagram with bottom sequence 2,3,3,2,1 and top sequence 2,3,2,3,1, namely:

Note that I read from right to left. The bottom sequence of numbers is the position of the strings. The degree of the diagram is (q^2 + 1)^2/q.permuation

All other outputs are read similarly. Note that we can get more than one diagram, for example

   klr_cyclotomic_dimension(['B',3],[2,3], [2,3,3,2,1], [2,3,2,3,1])

gives

   ({(q^2 + 1/q^2)*q^4: [(0,2,3)],
  (q^2 + 1/q^2 + 1)*(q^2 + 1/q^2)*q^4: [(0,2,1,3)]},
 (q^4 + 1)*(q^2 + 1)^2)

Thus, there are two relevant diagrams, one determined by the permutation (0,2,3) and the other by (0,2,1,3).

What the code can do - more advanced

If we turn on verbose=True, then for example

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

we get

Subgroup of permutations = [(2,3), (0,2,3), (1,3,2), (0,2,1,3)]
N(1,t)-1: 0  2  0  0  1
N(w,t):   1  3  3  0  2
N(w,t):   1  1  1  1  2
N((0,2,3),t):   1  1  1  1  2
X((0,2,3)): (q^4 + 1)*q^2
N(w,t):   1  3  1  0  2
N(w,t):   1  3  1  1  2
N((0,2,1,3),t):   1  3  1  1  2
X((0,2,1,3)): (q^4 + 1)*(q^2 + q + 1)*(q^2 - q + 1)
(q^2 + 1/q^2)*q^4: (0,2,3)
(q^2 + 1/q^2 + 1)*(q^2 + 1/q^2)*q^4: (0,2,1,3)

({(q^2 + 1/q^2)*q^4: [(0,2,3)],
  (q^2 + 1/q^2 + 1)*(q^2 + 1/q^2)*q^4: [(0,2,1,3)]},
 (q^4 + 1)*(q^2 + 1)^2)

which is saying that the code tried four permutations, but only two are relevant. In this example there are no cancellations, but try for example

klr_cyclotomic_dimension(['F',4],[2,3], [2,3,3,2,1,2,3,3,2], [2,3,2,3,1,2,3,3,2], verbose=True)

to see cancellations.

Turning on the base, for example

   klr_cyclotomic_dimension(['B',3],[2,3], [3,3,2,1], [3,2,3,1], base=[2])

makes the computation faster, but might get the wrong result. In the above example we get

   ({}, 0)

which is wrong. We should get

   ({(q^2 + 1/q^2)*q^4: [(0,2,3)],
  (q^2 + 1/q^2 + 1)*(q^2 + 1/q^2)*q^4: [(0,2,1,3)]},
 (q^4 + 1)*(q^2 + 1)^2)

but since we excluded the first entry, the 0th position, the above permutations cannot appear, so we get zero as the result.

base is still useful for larger computations, but use with care.

Contact

If there are any questions, then please feel free to contact me: dtubbenhauer@gmail.com

Have fun playing with the code!