A new modular plethystic $\mathrm{SL}_2(\mathbb{F})$-isomorphism $\mathrm{Sym}^{N-1}E \otimes \bigwedge^{N+1} \mathrm{Sym}^{d+1}E \cong Δ^{(2,1^{N-1})} \mathrm{Sym}^d E$
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2024
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| author | Martinez, Alvaro L. Wildon, Mark |
| author_facet | Martinez, Alvaro L. Wildon, Mark |
| contents | Let $\mathbb{F}$ be a field and let $E$ be the natural representation of $\mathrm{SL}_2(\mathbb{F})$. Given a vector space $V$, let $Δ^{(2,1^{N-1})}V$ be the kernel of the multiplication map $\bigwedge^N V \otimes V \rightarrow \bigwedge^{N+1}V$. We construct an explicit $\mathrm{SL}_2(\mathbb{F})$-isomorphism $\mathrm{Sym}^{N-1}E \otimes \bigwedge^{N+1} \mathrm{Sym}^{d+1}E \cong Δ^{(2,1^{N-1})} \mathrm{Sym}^d E$. This $\mathrm{SL}_2(\mathbb{F})$-isomorphism is a modular lift of the $q$-binomial identity $q^{\frac{N(N-1)}{2}}[N]_q \binom{d+1}{N+1}_q = s_{(2,1^{N-1})}(1,q,\ldots, q^d)$, where $s_{(2,1^{N-1})}$ is the Schur function for the partition $(2,1^{N-1})$. This identity, which follows from our main theorem, implies the existence of an isomorphism when $\mathbb{F}$ is the field of complex numbers but it is notable, and not typical of the general case, that there is an explicit isomorphism defined in a uniform way for any field. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_04631 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A new modular plethystic $\mathrm{SL}_2(\mathbb{F})$-isomorphism $\mathrm{Sym}^{N-1}E \otimes \bigwedge^{N+1} \mathrm{Sym}^{d+1}E \cong Δ^{(2,1^{N-1})} \mathrm{Sym}^d E$ Martinez, Alvaro L. Wildon, Mark Representation Theory Combinatorics 20C20 (Primary) 05E05, 05E10, 17B10, 20G05 (Secondary) Let $\mathbb{F}$ be a field and let $E$ be the natural representation of $\mathrm{SL}_2(\mathbb{F})$. Given a vector space $V$, let $Δ^{(2,1^{N-1})}V$ be the kernel of the multiplication map $\bigwedge^N V \otimes V \rightarrow \bigwedge^{N+1}V$. We construct an explicit $\mathrm{SL}_2(\mathbb{F})$-isomorphism $\mathrm{Sym}^{N-1}E \otimes \bigwedge^{N+1} \mathrm{Sym}^{d+1}E \cong Δ^{(2,1^{N-1})} \mathrm{Sym}^d E$. This $\mathrm{SL}_2(\mathbb{F})$-isomorphism is a modular lift of the $q$-binomial identity $q^{\frac{N(N-1)}{2}}[N]_q \binom{d+1}{N+1}_q = s_{(2,1^{N-1})}(1,q,\ldots, q^d)$, where $s_{(2,1^{N-1})}$ is the Schur function for the partition $(2,1^{N-1})$. This identity, which follows from our main theorem, implies the existence of an isomorphism when $\mathbb{F}$ is the field of complex numbers but it is notable, and not typical of the general case, that there is an explicit isomorphism defined in a uniform way for any field. |
| title | A new modular plethystic $\mathrm{SL}_2(\mathbb{F})$-isomorphism $\mathrm{Sym}^{N-1}E \otimes \bigwedge^{N+1} \mathrm{Sym}^{d+1}E \cong Δ^{(2,1^{N-1})} \mathrm{Sym}^d E$ |
| topic | Representation Theory Combinatorics 20C20 (Primary) 05E05, 05E10, 17B10, 20G05 (Secondary) |
| url | https://arxiv.org/abs/2405.04631 |