Hook restriction coefficients
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909965682212864 |
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| author | Narayanan, Sridhar P. |
| author_facet | Narayanan, Sridhar P. |
| contents | The permutation matrices form a subgroup of $\text{GL}_n(\mathbb{C})$ that is isomorphic to the symmetric group $S_n$. Let $r_{μλ}$ denote the multiplicity of the irreducible representation $V_μ$ of $S_n$, corresponding to a partition $μ$ of $n$, in the restriction of an irreducible polynomial representation $W_λ(\mathbb{C})$ of $\text{GL}_n(\mathbb{C})$, corresponding to a partition $λ$ with at most $n$ parts. Finding a combinatorial interpretation for $r_{μλ}$ remains an open problem in algebraic combinatorics, called the \emph{restriction problem}.
We derive a new nonrecursive expression for a character polynomial called the \emph{Specht polynomial} and use it to find a combinatorial interpretation of $r_{μλ}$ when $λ$ is a hook-shaped partition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03443 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hook restriction coefficients Narayanan, Sridhar P. Combinatorics Representation Theory 05E10 (Primary), 20C15, 20C30 (Secondary) The permutation matrices form a subgroup of $\text{GL}_n(\mathbb{C})$ that is isomorphic to the symmetric group $S_n$. Let $r_{μλ}$ denote the multiplicity of the irreducible representation $V_μ$ of $S_n$, corresponding to a partition $μ$ of $n$, in the restriction of an irreducible polynomial representation $W_λ(\mathbb{C})$ of $\text{GL}_n(\mathbb{C})$, corresponding to a partition $λ$ with at most $n$ parts. Finding a combinatorial interpretation for $r_{μλ}$ remains an open problem in algebraic combinatorics, called the \emph{restriction problem}. We derive a new nonrecursive expression for a character polynomial called the \emph{Specht polynomial} and use it to find a combinatorial interpretation of $r_{μλ}$ when $λ$ is a hook-shaped partition. |
| title | Hook restriction coefficients |
| topic | Combinatorics Representation Theory 05E10 (Primary), 20C15, 20C30 (Secondary) |
| url | https://arxiv.org/abs/2403.03443 |