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1
vote
0answers
93 views

Representation of Permutation group: What is the isormorphism between the equivalent descriptions, (1)using schur functions and (2)abstract tensors

I assume that the two ways of describing the representation of the permutation group using abstract tensors (perhaps more naive but widely used in physics) and schur functions are equivalent. If ...
2
votes
3answers
243 views

Is the reduced plethysm (restricted to 2-columns in Young tableaux) of this Schur funtion known $\mathbb S_{3^1}(\mathbb S_{1^p})$?

I am working on a physical problem, where I need to compute the "reduced plethysm" that is all the irreducibles characterised by the Young tableaux of 2 columns or less. The plethysm problem I want to ...
5
votes
1answer
165 views

A formula on Kronecker coefficients

Accidentally, I proved the following formula for the Kronecker coefficients using some obscure method. $$g_{lm^n,mn^l}^{m^{nl}}=1,\ \forall l,m,n\in\mathbb{N},$$ where $n^m$ is the rectangle ...
4
votes
2answers
296 views

Symmetric powers of Schur polynomials

I apologize if this question is trivial, but a couple of days of searching for necessary routines have led me here. Does there exist software to compute symmetric powers of Schur polynomials? ...
3
votes
3answers
416 views

Decomposition into irreducibles of symmetric powers of irreps.

Suppose I have an irreducible representation of a simple Lie algebra, say $\mathfrak{sl}(n)$ or $\mathfrak{so}(n)$ i.e., $A$ and $D$ type. Given such a representation, $\Gamma_\lambda$, indexed by its ...
8
votes
1answer
272 views

Plethysm of $\mathrm{QSym}$ into $\mathrm{QSym}$: can it be defined?

I will denote by $\Lambda$ the ring of symmetric functions, and by $\mathrm{QSym}$ the ring of quasisymmetric functions (both in infinitely many variables $x_1$, $x_2$, $x_3$, ..., both over $\mathbb ...
4
votes
1answer
349 views

Known decomposition of $\bigwedge^k Sym^d \mathbb C^n$ in special cases?

Let $V = \mathbb C^n$. Consider the plethysm $\bigwedge^k Sym^d V$ as a representation of $GL(V)$. In what special cases (e.g., for what $k$, $d$, and $n$) is this representation's decomposition into ...
11
votes
1answer
446 views

A question on invariant theory of $GL_n(\mathbb{C})$.

Let $\rho$ denote the irreducible algebraic representation of $GL_n(\mathbb{C})$ with the highest weight $(2,2,\underset{n-2}{\underbrace{0,\dots,0}})$. Let $k\leq n/2$ be a non-negative integer. ...
6
votes
1answer
251 views

Is a certain symmetric power reprsentation of GL(m) cyclically generated

Let $V_m$ be the $m$-dimensional complex vector space with basis $\{e_1, \dots, e_m\}$ and let $i\leq m$. Consider the element ${v}_0^i \in S^i(S^m(V_m))$, where ${v}_0$ is the element $e_1\dots e_m ...
8
votes
1answer
647 views

Symmetric tensor product of bosonic/fermionic Hilbert space

Consider two representation of the group $SU(n)$: $Sym^k(\mathbb{C}^n)$ and $\wedge^k\mathbb{C}^n$ ($k\leq n$) and take their symmetric tensor products: $Sym^2(Sym^k(\mathbb{C}^n))$, ...
5
votes
3answers
384 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 ...
6
votes
2answers
860 views

Is there a quantum Hermite reciprocity?

It is well known that there is an isomorphism of $SL_2=SL(V)$ representations $$ Sym^n(Sym^m(V))\simeq Sym^m(Sym^n(V)) $$ called Hermite reciprocity (discovered in 1854). My question is: Is there ...