# A Maple package for Kronecker coefficients ¶

`Kronecker`

is a Maple package for computing Kronecker coefficients g(λ,μ,ν). It implements a highly optimized version of the algorithm proposed by Baldoni-Vergne-Walter (2017). `Kronecker`

works just as well symbolically, computing *dilated* or *stretched* Kronecker coefficients g(kλ,kμ,kν) and even the entire *quasipolynomial* on a polyhedral chamber of (λ,μ,ν)’s.

# Getting started¶

Download the zip file **here**, unpack it, and run the `GettingStarted.mw`

worksheet.
You should see the following code and output at the top of the file:

```
> restart:
read "Kronecker.mpl":
Kronecker v1.0 by Velleda Baldoni, Michèle Vergne, Michael Walter (see https://github.com/amsqi/kronecker for more information).
```

If you instead see an error message please let us now by opening a new bug report. The worksheet also contains the following illustrative examples demonstrating the functionality of the Kronecker package:

## Calculate individual coefficients¶

Pass a list of partitions [λ,μ,ν] to `Kronecker`

to compute the corresponding Kronecker coefficient g(λ,μ,ν):

```
> Kronecker([[131, 39, 19, 11], [110, 90], [120, 80]]);
70
> Kronecker([[15, 10, 9, 4, 3, 2], [21, 14, 8], [27, 16]]);
148
> Kronecker([[400000, 200000, 100000], [500000, 100000, 100000], [300000, 200000, 200000]]);
1596403568763802677125206373630515086
```

This last example is a branching problem for SL(9) and hence might take a little while to compute (around 6 minutes).

## Calculate dilated coefficients¶

By providing an additional symbolic parameter to `Kronecker`

, we can instead compute the dilated Kronecker coefficient, i.e., the quasipolynomial function p(k) = g(kλ,kμ,kν):

```
> Kronecker([[10, 6, 2], [10, 8], [11, 7]], k);
3/8 + (5/8)*(-1)^k + (3/2)*k + (7/4)*k^2
> Kronecker([[1, 1, 1, 1, 1, 1], [2, 2, 2], [3, 3]], k);
1
```

We can also probe individual cosets by using the `coset`

option:

```
> Kronecker([[1, 1, 1, 1], [1, 1, 1, 1], [2, 2]], k, coset = 0);
(1/6)*k + 1
> Kronecker([[1, 1, 1, 1], [1, 1, 1, 1], [2, 2]], k, coset = 1);
-1/6 + (1/6)*k
```

```
> Kronecker([[1, 1], [1, 1], [1, 1], [1, 1]], k, coset = 0);
(2/3)*k + (1/6)*k^2 + (1/72)*k^3 + 1
> Kronecker([[1, 1], [1, 1], [1, 1], [1, 1]], k, coset = 1);
5/18 + (13/24)*k + (1/6)*k^2 + (1/72)*k^3
```

The last example shows that the `Kronecker`

package also supports iterated Kronecked coefficients indexed by four or more partitions.

## Calculate quasipolynomials on a chamber¶

Lastly, by providing three symbolic variables we can compute the entire quasipolynomial on the corresponding chamber:

```
> Kronecker([[132, 38, 19, 11], [110, 90], [120, 80]], [lambda, mu, nu]);
3/4 + (1/8)*(-1)^(lambda[4]+mu[1]+lambda[2]+nu[1]) - (1/2)*lambda[4]*mu[1] - (1/2)*lambda[4]*lambda[2] + (1/4)*lambda[4]^2 + (1/2)*lambda[4]*nu[1] + (1/2)*lambda[2] + (1/2)*lambda[3] - lambda[4] + (1/2)*mu[1] - (1/2)*nu[1] - (1/4)*lambda[3]^2 - (1/2)*lambda[3]*nu[1] + (1/2)*lambda[3]*mu[1] + (1/2)*lambda[3]*lambda[2] + (1/8)*(-1)^(lambda[3]+mu[1]+lambda[2]+nu[1])
```

Sometimes, the chamber and hence the quasipolynomial are not uniquely determined, because the given partition triple sits on a wall between two chambers.
In other cases, such as in the following, the chamber is uniquely determined, but `Kronecker`

is not aware of it since the partition triple sits on the boundary of the Kirwan cone.
In either case the `forceperturb`

option can be used to go ahead and select a chamber:

```
> factor(Kronecker([[288, 192, 174, 120, 30, 6], [343, 270, 197], [654, 156]], [lambda, mu, nu], forceperturb = true));
1/5040 * (-nu[1]+7+lambda[1]+lambda[2]+lambda[3]) * (-nu[1]+6+lambda[3]+lambda[2]+lambda[1]) * (-nu[1]+5+lambda[3]+lambda[2]+lambda[1]) * (-nu[1]+4+lambda[3]+lambda[2]+lambda[1]) * (-nu[1]+3+lambda[3]+lambda[2]+lambda[1]) * (-nu[1]+2+lambda[3]+lambda[2]+lambda[1]) * (-nu[1]+1+lambda[3]+lambda[2]+lambda[1]) * (lambda[1]+lambda[2]+lambda[4]+lambda[5]-mu[1]-mu[2]+1)
```

# Attribution¶

If you find this software useful in your research please consider citing our paper:

```
@article{kronecker,
title={Computation of dilated Kronecker coefficients},
author={Baldoni, Velleda and Vergne, Mich{\`e}le},
journal={Journal of Symbolic Computation},
year={2017},
doi={10.1016/j.jsc.2017.03.005},
eprint={1601.04325},
note={In press. With an appendix by Michael Walter. Software available at \url{https://amsqi.github.io/kronecker/}.},
}
```

# See also¶

barvikron: a Python package for efficiently computing Kronecker coefficients (using Barvinok’s algorithm to evaluate characters rather than the iterated residues used here)

LiE: a computer algebra system for reductive Lie group computations

SageMath: a computer algebra system which includes support for symmetric functions