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Identifier
Values
=>
Cc0002;cc-rep
[1]=>1 [2]=>0 [1,1]=>0 [3]=>0 [2,1]=>1 [1,1,1]=>0 [4]=>0 [3,1]=>0 [2,2]=>1 [2,1,1]=>0 [1,1,1,1]=>0 [5]=>0 [4,1]=>0 [3,2]=>0 [3,1,1]=>1 [2,2,1]=>0 [2,1,1,1]=>0 [1,1,1,1,1]=>0 [6]=>0 [5,1]=>0 [4,2]=>0 [4,1,1]=>0 [3,3]=>0 [3,2,1]=>1 [3,1,1,1]=>0 [2,2,2]=>0 [2,2,1,1]=>0 [2,1,1,1,1]=>0 [1,1,1,1,1,1]=>0 [7]=>0 [6,1]=>0 [5,2]=>0 [5,1,1]=>0 [4,3]=>0 [4,2,1]=>0 [4,1,1,1]=>1 [3,3,1]=>0 [3,2,2]=>0 [3,2,1,1]=>0 [3,1,1,1,1]=>0 [2,2,2,1]=>0 [2,2,1,1,1]=>0 [2,1,1,1,1,1]=>0 [1,1,1,1,1,1,1]=>0 [8]=>0 [7,1]=>0 [6,2]=>0 [6,1,1]=>0 [5,3]=>0 [5,2,1]=>0 [5,1,1,1]=>0 [4,4]=>0 [4,3,1]=>0 [4,2,2]=>0 [4,2,1,1]=>1 [4,1,1,1,1]=>0 [3,3,2]=>1 [3,3,1,1]=>0 [3,2,2,1]=>0 [3,2,1,1,1]=>0 [3,1,1,1,1,1]=>0 [2,2,2,2]=>0 [2,2,2,1,1]=>0 [2,2,1,1,1,1]=>0 [2,1,1,1,1,1,1]=>0 [1,1,1,1,1,1,1,1]=>0 [9]=>0 [8,1]=>0 [7,2]=>0 [7,1,1]=>0 [6,3]=>0 [6,2,1]=>0 [6,1,1,1]=>0 [5,4]=>0 [5,3,1]=>0 [5,2,2]=>0 [5,2,1,1]=>0 [5,1,1,1,1]=>1 [4,4,1]=>0 [4,3,2]=>0 [4,3,1,1]=>0 [4,2,2,1]=>0 [4,2,1,1,1]=>0 [4,1,1,1,1,1]=>0 [3,3,3]=>1 [3,3,2,1]=>0 [3,3,1,1,1]=>0 [3,2,2,2]=>0 [3,2,2,1,1]=>0 [3,2,1,1,1,1]=>0 [3,1,1,1,1,1,1]=>0 [2,2,2,2,1]=>0 [2,2,2,1,1,1]=>0 [2,2,1,1,1,1,1]=>0 [2,1,1,1,1,1,1,1]=>0 [1,1,1,1,1,1,1,1,1]=>0 [10]=>0 [9,1]=>0 [8,2]=>0 [8,1,1]=>0 [7,3]=>0 [7,2,1]=>0 [7,1,1,1]=>0 [6,4]=>0 [6,3,1]=>0 [6,2,2]=>0 [6,2,1,1]=>0 [6,1,1,1,1]=>0 [5,5]=>0 [5,4,1]=>0 [5,3,2]=>0 [5,3,1,1]=>0 [5,2,2,1]=>0 [5,2,1,1,1]=>1 [5,1,1,1,1,1]=>0 [4,4,2]=>0 [4,4,1,1]=>0 [4,3,3]=>0 [4,3,2,1]=>1 [4,3,1,1,1]=>0 [4,2,2,2]=>0 [4,2,2,1,1]=>0 [4,2,1,1,1,1]=>0 [4,1,1,1,1,1,1]=>0 [3,3,3,1]=>0 [3,3,2,2]=>0 [3,3,2,1,1]=>0 [3,3,1,1,1,1]=>0 [3,2,2,2,1]=>0 [3,2,2,1,1,1]=>0 [3,2,1,1,1,1,1]=>0 [3,1,1,1,1,1,1,1]=>0 [2,2,2,2,2]=>0 [2,2,2,2,1,1]=>0 [2,2,2,1,1,1,1]=>0 [2,2,1,1,1,1,1,1]=>0 [2,1,1,1,1,1,1,1,1]=>0 [1,1,1,1,1,1,1,1,1,1]=>0 [11]=>0 [10,1]=>0 [9,2]=>0 [9,1,1]=>0 [8,3]=>0 [8,2,1]=>0 [8,1,1,1]=>0 [7,4]=>0 [7,3,1]=>0 [7,2,2]=>0 [7,2,1,1]=>0 [7,1,1,1,1]=>0 [6,5]=>0 [6,4,1]=>0 [6,3,2]=>0 [6,3,1,1]=>0 [6,2,2,1]=>0 [6,2,1,1,1]=>0 [6,1,1,1,1,1]=>1 [5,5,1]=>0 [5,4,2]=>0 [5,4,1,1]=>0 [5,3,3]=>0 [5,3,2,1]=>0 [5,3,1,1,1]=>0 [5,2,2,2]=>0 [5,2,2,1,1]=>0 [5,2,1,1,1,1]=>0 [5,1,1,1,1,1,1]=>0 [4,4,3]=>0 [4,4,2,1]=>0 [4,4,1,1,1]=>0 [4,3,3,1]=>1 [4,3,2,2]=>0 [4,3,2,1,1]=>0 [4,3,1,1,1,1]=>0 [4,2,2,2,1]=>0 [4,2,2,1,1,1]=>0 [4,2,1,1,1,1,1]=>0 [4,1,1,1,1,1,1,1]=>0 [3,3,3,2]=>0 [3,3,3,1,1]=>0 [3,3,2,2,1]=>0 [3,3,2,1,1,1]=>0 [3,3,1,1,1,1,1]=>0 [3,2,2,2,2]=>0 [3,2,2,2,1,1]=>0 [3,2,2,1,1,1,1]=>0 [3,2,1,1,1,1,1,1]=>0 [3,1,1,1,1,1,1,1,1]=>0 [2,2,2,2,2,1]=>0 [2,2,2,2,1,1,1]=>0 [2,2,2,1,1,1,1,1]=>0 [2,2,1,1,1,1,1,1,1]=>0 [2,1,1,1,1,1,1,1,1,1]=>0 [1,1,1,1,1,1,1,1,1,1,1]=>0 [12]=>0 [11,1]=>0 [10,2]=>0 [10,1,1]=>0 [9,3]=>0 [9,2,1]=>0 [9,1,1,1]=>0 [8,4]=>0 [8,3,1]=>0 [8,2,2]=>0 [8,2,1,1]=>0 [8,1,1,1,1]=>0 [7,5]=>0 [7,4,1]=>0 [7,3,2]=>0 [7,3,1,1]=>0 [7,2,2,1]=>0 [7,2,1,1,1]=>0 [7,1,1,1,1,1]=>0 [6,6]=>0 [6,5,1]=>0 [6,4,2]=>0 [6,4,1,1]=>0 [6,3,3]=>0 [6,3,2,1]=>0 [6,3,1,1,1]=>0 [6,2,2,2]=>0 [6,2,2,1,1]=>0 [6,2,1,1,1,1]=>1 [6,1,1,1,1,1,1]=>0 [5,5,2]=>0 [5,5,1,1]=>0 [5,4,3]=>0 [5,4,2,1]=>0 [5,4,1,1,1]=>0 [5,3,3,1]=>0 [5,3,2,2]=>0 [5,3,2,1,1]=>1 [5,3,1,1,1,1]=>0 [5,2,2,2,1]=>0 [5,2,2,1,1,1]=>0 [5,2,1,1,1,1,1]=>0 [5,1,1,1,1,1,1,1]=>0 [4,4,4]=>0 [4,4,3,1]=>0 [4,4,2,2]=>1 [4,4,2,1,1]=>0 [4,4,1,1,1,1]=>0 [4,3,3,2]=>0 [4,3,3,1,1]=>0 [4,3,2,2,1]=>0 [4,3,2,1,1,1]=>0 [4,3,1,1,1,1,1]=>0 [4,2,2,2,2]=>0 [4,2,2,2,1,1]=>0 [4,2,2,1,1,1,1]=>0 [4,2,1,1,1,1,1,1]=>0 [4,1,1,1,1,1,1,1,1]=>0 [3,3,3,3]=>0 [3,3,3,2,1]=>0 [3,3,3,1,1,1]=>0 [3,3,2,2,2]=>0 [3,3,2,2,1,1]=>0 [3,3,2,1,1,1,1]=>0 [3,3,1,1,1,1,1,1]=>0 [3,2,2,2,2,1]=>0 [3,2,2,2,1,1,1]=>0 [3,2,2,1,1,1,1,1]=>0 [3,2,1,1,1,1,1,1,1]=>0 [3,1,1,1,1,1,1,1,1,1]=>0 [2,2,2,2,2,2]=>0 [2,2,2,2,2,1,1]=>0 [2,2,2,2,1,1,1,1]=>0 [2,2,2,1,1,1,1,1,1]=>0 [2,2,1,1,1,1,1,1,1,1]=>0 [2,1,1,1,1,1,1,1,1,1,1]=>0 [1,1,1,1,1,1,1,1,1,1,1,1]=>0 [5,4,3,1]=>0 [5,4,2,2]=>0 [5,4,2,1,1]=>0 [5,3,3,2]=>0 [5,3,3,1,1]=>1 [5,3,2,2,1]=>0 [4,4,3,2]=>1 [4,4,3,1,1]=>0 [4,4,2,2,1]=>0 [4,3,3,2,1]=>0 [5,4,3,2]=>0 [5,4,3,1,1]=>0 [5,4,2,2,1]=>1 [5,3,3,2,1]=>0 [4,4,3,2,1]=>0 [5,4,3,2,1]=>1
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Description
The multiplicity of the sign representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{1^n}$, for $\lambda\vdash n$. It equals $1$ if and only if $\lambda$ is self-conjugate.
Code
from sage.libs.symmetrica.symmetrica import charvalue_symmetrica as chv
def kronecker_coefficient(*partns):
    if partns == ():
        return 1
    else:
        return sum(mul(chv(la,mu) for la in partns)/mu.centralizer_size() for mu in Partitions(sum(partns[0])))

def statistic(la):
    if la.size():
        return kronecker_coefficient(la,la,[1]*la.size())
    return 1
Created
Mar 17, 2018 at 11:57 by Martin Rubey
Updated
Jun 25, 2021 at 09:36 by Martin Rubey