Elementary methods for splitting representations of Rook monoids: a gentle introduction to groupoids
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929570070921216 |
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| author | Duchamp, Gérard Henry Edmond Geloun, Joseph Ben Tollu, Christophe |
| author_facet | Duchamp, Gérard Henry Edmond Geloun, Joseph Ben Tollu, Christophe |
| contents | We show that the algebra of the coloured rook monoid $R_n^{(r)}$, {\em i.e.} the monoid of $n \times n$ matrices with at most one non-zero entry (an $r$-th root of unity) in each column and row, is the algebra of a finite groupoid, thus is endowed with a $C^*$-algebra structure. This new perspective uncovers the representation theory of these monoid algebras by making manifest their decomposition in irreducible modules. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_23731 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Elementary methods for splitting representations of Rook monoids: a gentle introduction to groupoids Duchamp, Gérard Henry Edmond Geloun, Joseph Ben Tollu, Christophe Discrete Mathematics We show that the algebra of the coloured rook monoid $R_n^{(r)}$, {\em i.e.} the monoid of $n \times n$ matrices with at most one non-zero entry (an $r$-th root of unity) in each column and row, is the algebra of a finite groupoid, thus is endowed with a $C^*$-algebra structure. This new perspective uncovers the representation theory of these monoid algebras by making manifest their decomposition in irreducible modules. |
| title | Elementary methods for splitting representations of Rook monoids: a gentle introduction to groupoids |
| topic | Discrete Mathematics |
| url | https://arxiv.org/abs/2410.23731 |