Questions tagged [schur-functors]
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35 questions
3
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Are there infinitely many $m$ such that $S^m V$ appears in $S^{(2,1)}(S^m V)$ for $\mathrm{Sp}(V)$?
Let $V$ be the standard $2g$-dimensional representation of $\mathrm{Sp}(V)$ (with $g \ge 1$), and for each $m \ge 1$ let $S^m V$ denote the $m$-th symmetric power.
Consider the Schur functor $S^{(2,1)}...
- 3,416
2
votes
0
answers
86
views
How can I define an invariant inner product on a Schur module?
As the title. In general, given a partition or say, a Young diagram $\lambda$ and a linear space $V$ of dimension $n$, we can construct a Schur module corresponding to the irreducible $GL(n)$-...
- 51
3
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152
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Why Schur modules are images of Young symmetrizers over characteristic zero?
There are two versions of Schur functors. One is defined by
Akin, Kaan; Buchsbaum, David A.; Weyman, Jerzy, Schur functors and Schur complexes, Adv. Math. 44, 207-278 (1982). ZBL0497.15020.
which ...
- 51
7
votes
1
answer
453
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Coefficients when rewriting the Hook-Content polynomials in terms of binomial polynomials
Let $\lambda$ be a partition, $S_\lambda$ the Schur functor attached to $\lambda$, and let $p_\lambda(t)$ be the polynomial determined by the condition that $\dim S_\lambda(k^n) = p_\lambda(n)$ for ...
- 28.7k
2
votes
0
answers
163
views
Rank and determinant of the image of a vector bundle after applying a Schur functor?
Let $\mathcal{E}$ be a vector bundle of rank $r$ and degree $d$ over some smooth projective variety $X$. Furthermore, let $\lambda$ be a partition of $n$. We apply the $\lambda$-th Schur functor to $\...
- 21
1
vote
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157
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The S-module Ass is same as the composite of Com and Lie
It has been cited in several places (eg. https://arxiv.org/pdf/1912.05519.pdf) that the S-module Ass is isomorphic to the composite of the S-modules Com and Lie. Is there a reference which gives the ...
- 101
4
votes
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250
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Relationship between $\mathbb{S}^{\nu}V \otimes \mathbb{S}^{\lambda}(V^{*})$ and $\mathbb{S}^{\nu / \lambda}V$
For partition $\mu$ let $\mathbb{S}^{\mu}V = V^{\otimes \mu} \cdot c_{\mu}$, where $c_{\mu}$ is the Young symmetrizer. I'm trying to prove that $\mathbb{S}^{\nu / \lambda}V$ is the polynomial part of $...
- 59
6
votes
1
answer
529
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Yoneda map for a composition of a representable functor and an arbitrary functor
Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $T : \mathcal{C} \rightarrow \mathcal{D}$ be a functor. Suppose that $F : \mathcal{D}^\mathrm{op} \rightarrow \mathrm{Set}$ is a functor. (So ...
- 12.2k
2
votes
0
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99
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Schur Bundle of Smooth Manifold
I've seen hints at the following result:
Let $M$ be a 3-dimensional manifold and let $T := T^*M$ be the cotangent bundle. By Schur-Weyl Duality, the 3rd tensor product can be written as follows:
$$T^...
- 941
3
votes
0
answers
271
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Decomposing Schur functor applied to a tensor product
I want to compute
$$
S^{2,2,\dots,2,1}(\mathbb C^{2m-1} \otimes W)^{SL(2m-1)}
$$
Here $m$ numbers should appear in the superscipt of the Schur functor, and the last superscript means to take $SL(2m-1)$...
- 1,559
7
votes
0
answers
3k
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The symmetric power of a tensor product
In the representation theory, if $S^{\lambda}(V)$ is the irreductible representation of $\text{GL}(V)$ associated to a partition $\lambda \vdash n$ (in perticular, $S^n(V)$ is the $n^{\text{th}}$ ...
- 941
6
votes
0
answers
150
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Natural maps between Schur functors: understanding the image
Let $V$ be a finite dimensional representation of symmetric group $\mathbb{S}_n.$ Consider a natural map
$$\pi \colon \Lambda^2 V \otimes \Lambda^2 V \longrightarrow \Lambda^4 V.$$
Let $[\Lambda^2 V]...
- 4,356
2
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0
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303
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The first non-trivial Schur functor [closed]
I am trying to understand the Schur functor $S^{(2,1)}$. Let's try on a vector space $V$ of dimension 3. The general definition is :
$S^{\lambda}V = V^{\otimes n} \otimes_{S_n} V^{\lambda}$
where $V^...
- 941
5
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1k
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Internal tensor product of strict polynomial functors: is there a more explicit definition?
In the paper Henning Krause, Koszul, Ringel, and Serre duality for strict polynomial functors, arXiv:1203.0311v4, Krause defines something that he calls an "internal tensor product" on the category of ...
- 35.9k
4
votes
0
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872
views
Categorifying the Cauchy kernel as a filtration of $\operatorname*{Sym}\left( F\otimes G\right) $ over any commutative ring
Question 1 (short version). Let $R$ be a commutative ring with unity. Let
$F$ and $G$ be two $R$-modules. Let $n\in\mathbb{N}$. Is it true that the
$n$-th symmetric power $\operatorname*{Sym}\...
- 35.9k