Questions tagged [schur-functors]
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33
questions
48
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How to constructively/combinatorially prove Schur-Weyl duality?
How is Schur-Weyl duality (specifically, the fact that the actions of the group ring $\mathbb{K}\left[ S_{n}\right] $ and the monoid ring
$\mathbb{K}\left[ \left( \operatorname*{End}V,\cdot\...
24
votes
2
answers
2k
views
Direct proof that the centralizer of $GL(V)$ acting on $V^{\otimes n}$ is spanned by $S_n$
Let $V$ be a finite dimensional vector space over a field of characteristic zero. Let $A$ be the space of maps in $\mathrm{End}(V^{\otimes n})$ which commute with the natural $GL(V)$ action. Clearly, ...
19
votes
3
answers
1k
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Is there "Schur-Weyl duality" for infinite dimensional unitary group?
To what extent does the relation between the diagonal representation of $SU(n)$ in $(\mathbb{C}^n)^{\otimes k}$ and representations of the symmetric group $S_k$ remain valid when instead of the group $...
12
votes
1
answer
644
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Schur functors generalization to "Jack", "Hall-Littlewood", "Macdonald" functors ?
Schur functors are functors from the category of vector spaces to itself.
If we take an operator $M: V->V$ and apply a Schur functor to it and then calculate trace $Tr(M^{\Lambda})$ we will get ...
11
votes
0
answers
410
views
Connection between Gelfand-Tsetlin basis and SSYT basis in Schur module
Consider an $n$-dimensional complex vector space $V$ with a chosen basis $e_1,\ldots,e_n$. This basis defines a Cartan decompostion of $GL(V)\cong GL_n$ and for an (integral dominant) highest weight $\...
9
votes
1
answer
1k
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Exact sequences of bundles on Grassmannians
We're looking for a large set of exact sequences of vector bundles on Grassmannians. Here's the set up:
$V$ and $Q$ are complex vector spaces of dimensions $d$ and $r$ respectively $(d\geq r)$, and ...
8
votes
1
answer
1k
views
A formula on Kronecker coefficients
Accidentally, I proved the following formula for the Kronecker coefficients using some obscure method.
$$g\bigl(\ell^{mn}, m^{\ell n},n^{\ell m}\bigr)=1,\ \forall l,m,n\in\mathbb{N},$$
where $n^m$ is ...
8
votes
2
answers
1k
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A basis for Schur functors
Suppose $V$ is a finite-dimensional vector space (over $\mathbb{C}$) and $\lambda$ is a partition of $n$ (not necessarily the dimension). Let $S^\lambda(V)=(V^{\otimes n})_\lambda$ be the $\lambda$'th ...
8
votes
1
answer
1k
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Details about plethysm
I'm currently working on plethysm, i.e. the character of the composition $S^\lambda(S^\mu(V))$ of the Schur functors $S^\lambda$ and $S^\mu$ on a complex vector space $V$. We note this character $s_\...
7
votes
1
answer
362
views
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 ...
7
votes
1
answer
448
views
Iterated Pieri's rule, Schur functors and intersection of subrepresentations
Let $\lambda$ and $\mu$ be two Young diagrams, such that $\lambda$ can be obtained from $\mu$ by extending one single column with additional $b$ boxes. Let $\Sigma^\lambda U$ and $\Sigma^\mu U$ denote ...
7
votes
1
answer
515
views
Question about decomposition of exterior product
In their paper "New lower bounds for the border rank of matrix multiplication", Landsberg and Ottaviani make use of the fact that
$$\tag{$\dagger$} {\textstyle\bigwedge}^p(V\otimes W) \cong \...
6
votes
1
answer
406
views
Yoneda map for a composition of a representable functor and an arbitrary functor
Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $T : \mathcal{C} \rightarrow \mathcal{D}$ be a functor. Suppose that $F : \mathcal{D}^\mathrm{op} \rightarrow \mathrm{Set}$ is a functor. (So ...
6
votes
0
answers
2k
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The symmetric power of a tensor product
In the representation theory, if $S^{\lambda}(V)$ is the irreductible representation of $\text{GL}(V)$ associated to a partition $\lambda \vdash n$ (in perticular, $S^n(V)$ is the $n^{\text{th}}$ ...
6
votes
0
answers
123
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Natural maps between Schur functors: understanding the image
Let $V$ be a finite dimensional representation of symmetric group $\mathbb{S}_n.$ Consider a natural map
$$\pi \colon \Lambda^2 V \otimes \Lambda^2 V \longrightarrow \Lambda^4 V.$$
Let $[\Lambda^2 V]...
5
votes
3
answers
609
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An isomorphism of 2-Schur modules
This is the little brother of question 68071: elementary, simple-looking and probably much easier to answer. Of course, it is just a small part of question 68071, as anybody with $\lambda$-rings ...
5
votes
0
answers
998
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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 ...
5
votes
0
answers
536
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Identities satisfied by the image of the Young symmetrizer
Consider a partition $\lambda=(r_1,\ldots,r_k)$ of an integer $n$ and the corresponding Young diagram with rows of length $r_1,\ldots,r_k$ (hence ordered in non-increasing order). Counting the column ...
5
votes
0
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261
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Explicit description of isomorphism when decomposing into irreps
I had a question which is slightly more general than this one on mathoverflow: I am looking for an explicit description of the isomorphism
$\mathbb S_\nu(V\otimes W) \cong \bigoplus C_{\lambda\mu\nu} \...
4
votes
1
answer
213
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A question on complex semisimple Lie groups and $(\mathbb{C}^2, \omega)$
Consider $(\mathbb{C}^2, \omega)$ where $\omega$ is a non-degenerate complex skew-symmetric bilinear form on $\mathbb{C}^2$. Let us write
$V = (\mathbb{C}^2, \omega)$
There are many spaces one can ...
4
votes
1
answer
882
views
Is there a generalization of Schur - Weyl duality and plethysm for direct product of special unitary groups?
Consider the semisimple compact group $K=SU(N_1)\times SU(N_2) \times \ldots \times SU(N_S)$ acting naturally on $\mathcal{H}=\mathcal{H}_1 \otimes \mathcal{H}_2 \otimes \ldots \otimes \mathcal{H}_S$, ...
4
votes
1
answer
612
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Can the projection (tensor algebra) -> (symmetric algebra) be forced to split in char. p by factoring out p-th powers?
Question 1 (the weak and simple statement, which, I think, already is wrong): Let $p$ be a prime. Let $k$ be a field with characteristic $p$.
For any $k$-vector space $V$, consider the canonical ...
4
votes
0
answers
206
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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 $...
4
votes
0
answers
838
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}\...
3
votes
1
answer
359
views
Young Symmetrizer and Exterior Products, such as $S_{(2,1)}V = Ker(\Lambda^2V \otimes V \to \Lambda^3V )$
Background: Young symmetrizer $c_\lambda$ gives an explicit description of Schur module $S_\lambda V$, which is also the kernel of maps between exterior products (as in Fulton & Harris).
Example:...
3
votes
0
answers
153
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Derived pushforward of a Schur functor, and bounded derived categories of Grassmannians
Consider Grassmanianns over fields of characteristic zero.
Let $i : Gr_{k-1,n} \rightarrow Gr_{k,n+1}$ be the `direct sum' map between Grassmannians. By universal property of Grassmannian, this map ...
3
votes
0
answers
188
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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)}
$$
Here $m$ numbers should appear in the superscipt of the Schur functor, and the last superscript means to take $SL(2m-1)$...
2
votes
1
answer
284
views
Symmetric invariants of a Schur Module
Let $V\cong\mathbb C^n$ be a complex vector space of dimension $n$. Let $\lambda\in\mathbb Z^r$ be a generalized integer partition $\lambda_1\ge\cdots\ge\lambda_r$ with $r\le n$. Denote by $\mathbb S_\...
2
votes
0
answers
57
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Rank and determinant of the image of a vector bundle after applying a Schur functor?
Let $\mathcal{E}$ be a vector bundle of rank $r$ and degree $d$ over some smooth projective variety $X$. Furthermore, let $\lambda$ be a partition of $n$. We apply the $\lambda$-th Schur functor to $\...
2
votes
0
answers
65
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Schur Bundle of Smooth Manifold
I've seen hints at the following result:
Let $M$ be a 3-dimensional manifold and let $T := T^*M$ be the cotangent bundle. By Schur-Weyl Duality, the 3rd tensor product can be written as follows:
$$T^...
2
votes
0
answers
195
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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^...
2
votes
0
answers
115
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Is the Symplectic Schur algebra a 0-faithful cover of the Brauer algebra?
The symplectic Schur algebra, $Sp_{2n}$ and the Brauer algebra, $B_r(n)$, are in Schur-Weyl duality over an algebraically closed field of characteristic $p$ (this is due to Doty et. al.). The ...
1
vote
0
answers
122
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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 ...