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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2507.02539 |
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| _version_ | 1866913924189782016 |
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| author | Campbell, John M. |
| author_facet | Campbell, John M. |
| contents | Given a direct sum $A$ of full matrix algebras, if there is a combinatorial interpretation associated with both the dimension of $A$ and the dimensions of the irreducible $A$-modules, then this can be thought of as providing an analogue of the famous Frobenius-Young identity $n! = \sum_{λ\vdash n} ( f^λ )^{2}$ derived from the semisimple structure of the symmetric group algebra $\mathbb{C}S_{n}$, letting $f^λ$ denote the number of Young tableaux of partition shape $λ\vdash n$. By letting $g^α$ denote the number of standard immaculate tableaux of composition shape $α\vDash n$, we construct an algebra $\mathbb{C}\mathcal{I}_{n}$ with a semisimple structure such that $\dim \mathbb{C}\mathcal{I}_{n} = \sum_{α\vDash n} (g^α)^{2}$ and such that $\mathbb{C}\mathcal{I}_{n} $ contains an isomorphic copy of $\mathbb{C}S_{n}$. We bijectively prove a recurrence for $\dim \mathbb{C}\mathcal{I}_{n}$ so as to construct a basis of $\mathbb{C}\mathcal{I}_{n}$ indexed by permutation-like objects that we refer to as immacutations. We form a basis $\mathcal{B}_{n}$ of $\mathbb{C}\mathcal{I}_{n}$ such that $\mathbb{C} \mathcal{B}_n$ has the structure of a monoid algebra in such a way so that $\mathcal{B}_n$ is closed under the multiplicative operation of $\mathbb{C} \mathcal{I}_n$, yielding a monoid structure on the set of order-$n$ immacutations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02539 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Semisimple algebras related to immaculate tableaux Campbell, John M. Combinatorics 05E10 Given a direct sum $A$ of full matrix algebras, if there is a combinatorial interpretation associated with both the dimension of $A$ and the dimensions of the irreducible $A$-modules, then this can be thought of as providing an analogue of the famous Frobenius-Young identity $n! = \sum_{λ\vdash n} ( f^λ )^{2}$ derived from the semisimple structure of the symmetric group algebra $\mathbb{C}S_{n}$, letting $f^λ$ denote the number of Young tableaux of partition shape $λ\vdash n$. By letting $g^α$ denote the number of standard immaculate tableaux of composition shape $α\vDash n$, we construct an algebra $\mathbb{C}\mathcal{I}_{n}$ with a semisimple structure such that $\dim \mathbb{C}\mathcal{I}_{n} = \sum_{α\vDash n} (g^α)^{2}$ and such that $\mathbb{C}\mathcal{I}_{n} $ contains an isomorphic copy of $\mathbb{C}S_{n}$. We bijectively prove a recurrence for $\dim \mathbb{C}\mathcal{I}_{n}$ so as to construct a basis of $\mathbb{C}\mathcal{I}_{n}$ indexed by permutation-like objects that we refer to as immacutations. We form a basis $\mathcal{B}_{n}$ of $\mathbb{C}\mathcal{I}_{n}$ such that $\mathbb{C} \mathcal{B}_n$ has the structure of a monoid algebra in such a way so that $\mathcal{B}_n$ is closed under the multiplicative operation of $\mathbb{C} \mathcal{I}_n$, yielding a monoid structure on the set of order-$n$ immacutations. |
| title | Semisimple algebras related to immaculate tableaux |
| topic | Combinatorics 05E10 |
| url | https://arxiv.org/abs/2507.02539 |