Questions tagged [schur-functors]

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2
votes
0answers
41 views

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
0answers
43 views

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 $SL(2m-1)$...
6
votes
0answers
392 views

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
0answers
101 views

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]...
2
votes
0answers
131 views

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^...
5
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0answers
646 views

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 ...
4
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0answers
747 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}\...
11
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0answers
298 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 $\...
7
votes
1answer
351 views

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_\...
39
votes
3answers
2k views

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\...
4
votes
1answer
142 views

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 ...
3
votes
1answer
242 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:...
4
votes
0answers
330 views

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 ...
0
votes
0answers
214 views

representations of the special orthogonal group

Consider an $N$-dimensional (algebraic) representation $r$ of the special orthogonal group $SO_m$ over the rational numbers $Q$. Is it true that there exists a representation $\varphi \colon GL_m \to ...
2
votes
1answer
246 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_\...
8
votes
1answer
788 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 ...
2
votes
0answers
105 views

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 ...
7
votes
1answer
375 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 ...
5
votes
0answers
243 views

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} \...
6
votes
2answers
851 views

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 ...
11
votes
1answer
524 views

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 ...
7
votes
1answer
373 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 \...
4
votes
1answer
681 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 $\mathcal{H}=\mathcal{H}_1 \otimes \mathcal{H}_2 \otimes \ldots \otimes \mathcal{H}_S$, ...
23
votes
2answers
1k 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, ...
17
votes
3answers
1k views

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 $...
5
votes
3answers
525 views

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 ...
9
votes
1answer
1k views

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 ...
4
votes
1answer
475 views

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 ...