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Identifier

Mp00186: Skew partitions —dominating partition⟶ Integer partitions

Images

[[1],[]] => [1]

[[2],[]] => [2]

[[1,1],[]] => [1,1]

[[2,1],[1]] => [2]

[[3],[]] => [3]

[[2,1],[]] => [2,1]

[[3,1],[1]] => [3]

[[2,2],[1]] => [2,1]

[[3,2],[2]] => [3]

[[1,1,1],[]] => [1,1,1]

[[2,2,1],[1,1]] => [2,1]

[[2,1,1],[1]] => [2,1]

[[3,2,1],[2,1]] => [3]

[[4],[]] => [4]

[[3,1],[]] => [3,1]

[[4,1],[1]] => [4]

[[2,2],[]] => [2,2]

[[3,2],[1]] => [3,1]

[[4,2],[2]] => [4]

[[2,1,1],[]] => [2,1,1]

[[3,2,1],[1,1]] => [3,1]

[[3,1,1],[1]] => [3,1]

[[4,2,1],[2,1]] => [4]

[[3,3],[2]] => [3,1]

[[4,3],[3]] => [4]

[[2,2,1],[1]] => [2,2]

[[3,3,1],[2,1]] => [3,1]

[[3,2,1],[2]] => [3,1]

[[4,3,1],[3,1]] => [4]

[[2,2,2],[1,1]] => [2,1,1]

[[3,3,2],[2,2]] => [3,1]

[[3,2,2],[2,1]] => [3,1]

[[4,3,2],[3,2]] => [4]

[[1,1,1,1],[]] => [1,1,1,1]

[[2,2,2,1],[1,1,1]] => [2,1,1]

[[2,2,1,1],[1,1]] => [2,2]

[[3,3,2,1],[2,2,1]] => [3,1]

[[2,1,1,1],[1]] => [2,1,1]

[[3,2,2,1],[2,1,1]] => [3,1]

[[3,2,1,1],[2,1]] => [3,1]

[[4,3,2,1],[3,2,1]] => [4]

[[5],[]] => [5]

[[4,1],[]] => [4,1]

[[5,1],[1]] => [5]

[[3,2],[]] => [3,2]

[[4,2],[1]] => [4,1]

[[5,2],[2]] => [5]

[[3,1,1],[]] => [3,1,1]

[[4,2,1],[1,1]] => [4,1]

[[4,1,1],[1]] => [4,1]

[[5,2,1],[2,1]] => [5]

[[3,3],[1]] => [3,2]

[[4,3],[2]] => [4,1]

[[5,3],[3]] => [5]

[[2,2,1],[]] => [2,2,1]

[[3,3,1],[1,1]] => [3,2]

[[3,2,1],[1]] => [3,2]

[[4,3,1],[2,1]] => [4,1]

[[4,2,1],[2]] => [4,1]

[[5,3,1],[3,1]] => [5]

[[3,2,2],[1,1]] => [3,1,1]

[[4,3,2],[2,2]] => [4,1]

[[4,2,2],[2,1]] => [4,1]

[[5,3,2],[3,2]] => [5]

[[2,1,1,1],[]] => [2,1,1,1]

[[3,2,2,1],[1,1,1]] => [3,1,1]

[[3,2,1,1],[1,1]] => [3,2]

[[4,3,2,1],[2,2,1]] => [4,1]

[[3,1,1,1],[1]] => [3,1,1]

[[4,2,2,1],[2,1,1]] => [4,1]

[[4,2,1,1],[2,1]] => [4,1]

[[5,3,2,1],[3,2,1]] => [5]

[[4,4],[3]] => [4,1]

[[5,4],[4]] => [5]

[[3,3,1],[2]] => [3,2]

[[4,4,1],[3,1]] => [4,1]

[[4,3,1],[3]] => [4,1]

[[5,4,1],[4,1]] => [5]

[[2,2,2],[1]] => [2,2,1]

[[3,3,2],[2,1]] => [3,2]

[[4,4,2],[3,2]] => [4,1]

[[3,2,2],[2]] => [3,2]

[[4,3,2],[3,1]] => [4,1]

[[5,4,2],[4,2]] => [5]

[[2,2,1,1],[1]] => [2,2,1]

[[3,3,2,1],[2,1,1]] => [3,2]

[[3,3,1,1],[2,1]] => [3,2]

[[4,4,2,1],[3,2,1]] => [4,1]

[[3,2,1,1],[2]] => [3,1,1]

[[4,3,2,1],[3,1,1]] => [4,1]

[[4,3,1,1],[3,1]] => [4,1]

[[5,4,2,1],[4,2,1]] => [5]

[[3,3,3],[2,2]] => [3,1,1]

[[4,4,3],[3,3]] => [4,1]

[[4,3,3],[3,2]] => [4,1]

[[5,4,3],[4,3]] => [5]

[[2,2,2,1],[1,1]] => [2,2,1]

[[3,3,3,1],[2,2,1]] => [3,1,1]

[[3,3,2,1],[2,2]] => [3,2]

[[4,4,3,1],[3,3,1]] => [4,1]

[[3,2,2,1],[2,1]] => [3,2]

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Description

The dominating partition in the Schur expansion.
Consider the expansion of the skew Schur function $s_{\lambda/\mu}=\sum_\nu c^\lambda_{\mu, \nu} s_\nu$ as a linear combination of straight Schur functions.
It is shown in [1] that the partitions $\nu$ with $c^\lambda_{\mu, \nu} > 0$ form a sublattice of the dominance order and that its top element is the conjugate of the partition formed by sorting the column lengths of $\lambda / \mu$ into decreasing order.
This map returns the largest partition $\nu$ in dominance order for which $c^\lambda_{\mu, \nu}$ is positive.
For example,
$$ s_{331/2} = s_{311} + s_{32}, $$
and the partition $32$ dominates $311$.

References

[1] Azenhas, O. The admissible interval for the invariant factors of a product of matrices MathSciNet:1712853

Properties

surjective, graded

Sage code

def mapping(la):
    return Partition(sorted(la.column_lengths(), reverse=True)).conjugate()

Weight

Created

Sep 27, 2020 at 14:56 by Martin Rubey

Updated

Sep 08, 2023 at 21:38 by Martin Rubey