Johnson graphs as slices of a hypercube and an algebra homomorphism from the universal Racah algebra into $U(\mathfrak{sl}_2)$
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929391981821952 |
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| author | Huang, Hau-Wen Wen, Chia-Yi |
| author_facet | Huang, Hau-Wen Wen, Chia-Yi |
| contents | From the viewpoint of Johnson graphs as slices of a hypercube, we derive a novel algebra homomorphism $\sharp$ from the universal Racah algebra $\Re$ into $U(\mathfrak{sl}_2)$. We use the Casimir elements of $\Re$ to describe the kernel of $\sharp$. By pulling back via $\sharp$ every $U(\mathfrak{sl}_2)$-module can be viewed as an $\Re$-module. We show that for any finite-dimensional $U(\mathfrak{sl}_2)$-module $V$, the $\Re$-module $V$ is completely reducible and three generators of $\Re$ act on every irreducible $\Re$-submodule of $V$ as a Leonard triple. In particular, Leonard triples can be constructed in terms of the second dual distance operator of the hypercube $H(D,2)$ and a decomposition of the second distance operator of $H(D,2)$ induced by Johnson graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_13455 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Johnson graphs as slices of a hypercube and an algebra homomorphism from the universal Racah algebra into $U(\mathfrak{sl}_2)$ Huang, Hau-Wen Wen, Chia-Yi Combinatorics Representation Theory 05E30, 16G30, 16S30, 33D45 From the viewpoint of Johnson graphs as slices of a hypercube, we derive a novel algebra homomorphism $\sharp$ from the universal Racah algebra $\Re$ into $U(\mathfrak{sl}_2)$. We use the Casimir elements of $\Re$ to describe the kernel of $\sharp$. By pulling back via $\sharp$ every $U(\mathfrak{sl}_2)$-module can be viewed as an $\Re$-module. We show that for any finite-dimensional $U(\mathfrak{sl}_2)$-module $V$, the $\Re$-module $V$ is completely reducible and three generators of $\Re$ act on every irreducible $\Re$-submodule of $V$ as a Leonard triple. In particular, Leonard triples can be constructed in terms of the second dual distance operator of the hypercube $H(D,2)$ and a decomposition of the second distance operator of $H(D,2)$ induced by Johnson graphs. |
| title | Johnson graphs as slices of a hypercube and an algebra homomorphism from the universal Racah algebra into $U(\mathfrak{sl}_2)$ |
| topic | Combinatorics Representation Theory 05E30, 16G30, 16S30, 33D45 |
| url | https://arxiv.org/abs/2406.13455 |