A new modular plethystic $\mathrm{SL}_2(\mathbb{F})$-isomorphism $\mathrm{Sym}^{N-1}E \otimes \bigwedge^{N+1} \mathrm{Sym}^{d+1}E \cong \Delta^{(2,1^{N-1})} \mathrm{Sym}^d E$
Abstract
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 $\Delta^{(2,1^{N-1})}V$ 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 $\mathrm{Sym}^{N-1}E \otimes \bigwedge^{N+1} \mathrm{Sym}^{d+1}E \cong \Delta^{(2,1^{N-1})} \mathrm{Sym}^d E$. 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.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2024
- DOI:
- 10.48550/arXiv.2405.04631
- arXiv:
- arXiv:2405.04631
- Bibcode:
- 2024arXiv240504631M
- Keywords:
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- Mathematics - Representation Theory;
- Mathematics - Combinatorics;
- 20C20 (Primary) 05E05;
- 05E10;
- 17B10;
- 20G05 (Secondary)
- E-Print:
- 16 pages, 1 figure