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### 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**

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**3**answers

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### 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 ...

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249 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 ...

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**1**answer

219 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 ...

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### 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. ...

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**1**answer

248 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 ...

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**1**answer

628 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: $Sym^2(Sym^k(\mathbb{C}^n))$, ...

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**3**answers

373 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 ...

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**2**answers

843 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 ...