Questions tagged [schur-functors]

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5 votes
3 answers
609 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 ...
3 votes
0 answers
153 views

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 ...
7 votes
1 answer
361 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 ...
2 votes
0 answers
57 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 $\...
1 vote
0 answers
122 views

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 ...
4 votes
0 answers
206 views

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 $...
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 ...
48 votes
4 answers
5k 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
1 answer
213 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 ...
2 votes
0 answers
65 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^...
3 votes
0 answers
188 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 $SL(2m-1)$...
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 ...
6 votes
0 answers
2k 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 $n^{\text{th}}$ ...
9 votes
1 answer
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 ...
6 votes
0 answers
123 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
0 answers
195 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 votes
0 answers
998 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 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}\...
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 $\...
8 votes
1 answer
1k 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_\...
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:...
5 votes
0 answers
536 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 ...
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_\...
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 ...
2 votes
0 answers
115 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 ...
5 votes
0 answers
261 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} \...
8 votes
2 answers
1k 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 ...
12 votes
1 answer
643 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
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 \...
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 $\mathcal{H}=\mathcal{H}_1 \otimes \mathcal{H}_2 \otimes \ldots \otimes \mathcal{H}_S$, ...
19 votes
3 answers
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 $...
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, ...
4 votes
1 answer
612 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 ...