Self-dual Grassmannian, Wronski map, and representations of $\mathfrak{gl}_N$, ${\mathfrak{sp}}_{2r}$, ${\mathfrak{so}}_{2r+1}$
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| Autores principales: | , , |
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| Formato: | Preprint |
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2017
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| _version_ | 1866915238185533440 |
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| author | Lu, Kang Mukhin, E. Varchenko, A. |
| author_facet | Lu, Kang Mukhin, E. Varchenko, A. |
| contents | We define a $\mathfrak{gl}_N$-stratification of the Grassmannian of $N$ planes $\mathrm{Gr}(N,d)$. The $\mathfrak{gl}_N$-stratification consists of strata $Ω_{\mathbfΛ}$ labeled by unordered sets $\mathbfΛ=(λ^{(1)},\dots,λ^{(n)})$ of nonzero partitions with at most $N$ parts, satisfying a condition depending on $d$, and such that $(\otimes_{i=1}^n V_{λ^{(i)}})^{\mathfrak{sl}_N}\ne 0$. Here $V_{λ^{(i)}}$ is the irreducible $\mathfrak{gl}_N$-module with highest weight $λ^{(i)}$. We show that the closure of a stratum $Ω_{\mathbfΛ}$ is the union of the strata $Ω_{\mathbfΞ}$, $\mathbfΞ=(ξ^{(1)},\dots,ξ^{(m)})$, such that there is a partition $\{I_1,\dots,I_m\}$ of $\{1,2,\dots,n\}$ with $ {\rm {Hom}}_{\mathfrak{gl}_N} (V_{ξ^{(i)}}, \otimes_{j\in I_i}V_{λ^{(j)}}\big)\neq 0$ for $i=1,\dots,m$. The $\mathfrak{gl}_N$-stratification of the Grassmannian agrees with the Wronski map.
We introduce and study the new object: the self-dual Grassmannian $\mathrm{sGr}(N,d)\subset \mathrm{Gr}(N,d)$. Our main result is a similar $\mathfrak{g}_N$-stratification of the self-dual Grassmannian governed by representation theory of the Lie algebra $\mathfrak {g}_{2r+1}:=\mathfrak{sp}_{2r}$ if $N=2r+1$ and of the Lie algebra $\mathfrak g_{2r}:=\mathfrak{so}_{2r+1}$ if $N=2r$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1705_02048 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Self-dual Grassmannian, Wronski map, and representations of $\mathfrak{gl}_N$, ${\mathfrak{sp}}_{2r}$, ${\mathfrak{so}}_{2r+1}$ Lu, Kang Mukhin, E. Varchenko, A. Quantum Algebra Algebraic Geometry Representation Theory We define a $\mathfrak{gl}_N$-stratification of the Grassmannian of $N$ planes $\mathrm{Gr}(N,d)$. The $\mathfrak{gl}_N$-stratification consists of strata $Ω_{\mathbfΛ}$ labeled by unordered sets $\mathbfΛ=(λ^{(1)},\dots,λ^{(n)})$ of nonzero partitions with at most $N$ parts, satisfying a condition depending on $d$, and such that $(\otimes_{i=1}^n V_{λ^{(i)}})^{\mathfrak{sl}_N}\ne 0$. Here $V_{λ^{(i)}}$ is the irreducible $\mathfrak{gl}_N$-module with highest weight $λ^{(i)}$. We show that the closure of a stratum $Ω_{\mathbfΛ}$ is the union of the strata $Ω_{\mathbfΞ}$, $\mathbfΞ=(ξ^{(1)},\dots,ξ^{(m)})$, such that there is a partition $\{I_1,\dots,I_m\}$ of $\{1,2,\dots,n\}$ with $ {\rm {Hom}}_{\mathfrak{gl}_N} (V_{ξ^{(i)}}, \otimes_{j\in I_i}V_{λ^{(j)}}\big)\neq 0$ for $i=1,\dots,m$. The $\mathfrak{gl}_N$-stratification of the Grassmannian agrees with the Wronski map. We introduce and study the new object: the self-dual Grassmannian $\mathrm{sGr}(N,d)\subset \mathrm{Gr}(N,d)$. Our main result is a similar $\mathfrak{g}_N$-stratification of the self-dual Grassmannian governed by representation theory of the Lie algebra $\mathfrak {g}_{2r+1}:=\mathfrak{sp}_{2r}$ if $N=2r+1$ and of the Lie algebra $\mathfrak g_{2r}:=\mathfrak{so}_{2r+1}$ if $N=2r$. |
| title | Self-dual Grassmannian, Wronski map, and representations of $\mathfrak{gl}_N$, ${\mathfrak{sp}}_{2r}$, ${\mathfrak{so}}_{2r+1}$ |
| topic | Quantum Algebra Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/1705.02048 |