# Questions tagged [plethysm]

The plethysm tag has no usage guidance.

32
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### Plethysm and wreath product

I am looking for a proof about the link between plethysm and wreath product. It is a well-known fact, being use extensively in many papers, but I can't find a good reference. Everything that follows ...

2
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0
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### A branching law involving 2-power exterior representations

Let $K=SU(n)$.
We take a maximal torus $T$ in $K$ and fixed a simple root system with fundamental weights $\eta_1,\dots,\eta_{n-1}$.
For $\mu$ a dominant weight of $K$, we denote by $(\tau_\mu,V_\mu)$ ...

8
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1
answer

352
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### Interaction of plethysm with other operations

The plethysm $s_{\nu}[s_{\mu}]$ of two symmetric functions is the character of the composition of Schur functors $S^{\nu}(S^{\mu}(V))$. We know that this operation is linear and multiplicative in its ...

4
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1
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### Ideal of rational normal curve of degree $d$

Let $A$ consist of the columns of the $2\times (d+1)$ matrix
$$A=\begin{pmatrix}
d & d-1 & \cdots & 1&0\\
0 & 1 & \cdots & d-1 &d
\end{pmatrix}$$
Then consider the ...

4
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2
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### Invariants in the symmetric algebra of a module

Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra, and $V$ an irreducible finite-dimensional $\mathfrak{g}$-module. Then $\mathfrak{g}$ also acts on the symmetric algebra $S(V)$...

10
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3
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### What is known about the plethysm $\text{Sym}^d(\bigwedge^3 \mathbb{C}^6)$

What is known about the plethysm $\text{Sym}^d(\bigwedge^3 \mathbb{C}^6)$ as a representation of $\text{GL}(6)$? It is my understanding that this should be multiplicity-free. I tried computing it ...

2
votes

1
answer

325
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### Exterior powers of $Sym^p T$ over Gr(k,n)

Let G=Gr(k,n) the Grassmannian of $k$-dimensional subspaces of $\mathbb{C}^n$ and denote by $T$ the (rank $k$) tautological bundle over $G$, and by $Sym^p T$ its $p$-th symmetric power. Is there any ...

2
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1
answer

551
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### Homogeneous polynomials and symmetric binary forms

Let $f\in k[x_0,...,x_n]_d$ be a degree $d$ homogeneous polynomial in $n+1$ variables.
Is there a way to associate to $f$ a form $g(y_1,...,y_m)$ which is symmetric in the sets of binary variables $...

8
votes

1
answer

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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_\...

4
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1
answer

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### What is known about the decomposition of $Sym(Sym^3(V))$ into irreducibles?

The representation $\text{Sym}(\text{Sym}^3(V))$ of $\text{GL}(V)$ decomposes into a direct sum of $S_{\lambda}(V)$, where the $S_{\lambda}$ are Schur functors. What is know about this decomposition?
...

8
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### If $A$ is an algebra, $Sym^n(A)$ is an algebra. Where can I learn more about this algebra structure?

$\newcommand{\Vect}{\mathsf{Vect}}
\newcommand{\nats}{\mathbb{N}}
\newcommand{\Sym}{\mathrm{Sym}}
\newcommand{\Alg}{\mathsf{Alg}}
\newcommand{\CAlg}{\mathsf{CAlg}}
\newcommand{\Hom}{\mathrm{Hom}}$
Let ...

10
votes

1
answer

401
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### Super-plethysm?

Let $U$ be a representation of $S_m$ and $V$ a representation of $S_n$. Then the representation $\operatorname{Ind}_{S_m\wr S_n}^{S_{mn}}(U^{\otimes{n}}\otimes V)$ has a nice interpretation in terms ...

3
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0
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325
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### what is the link between plethysm in regular representation of the symmetric group and plethysm in Schur functions.

I am trying to understand first how one can define the plethysm say $s_\lambda \circ s_\mu$ as a module in the regular representation of the symmetric group.
1)How is it connected to the plethysms ...

2
votes

0
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657
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### What is the definition of plethysm in the representation theory of permutation groups

Let $s_\lambda \circ s_\mu$ be a plethysm. Here let $\lambda, \mu$ be $m,n$ box Young diagrams.
I have seen the definition of plethysms in symmetric functions. I would like to understand the ...

2
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0
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### Symmetric and antisymmetric powers of SU(2) representations [closed]

Recently, I took a course in representation theory at Imperial College, and on the first homework the questions were about certain sneaky relationships when it came to representations of SU(2).
...

4
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0
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### Efficiently computing (plethysm-like?)substitutions of symmetric functions

This is a rather technical question, it arose in connection of some calculations that I need to have better grasp of the question Formal group law over $\mathbb{F}_p$ and my own older one What is ...

1
vote

0
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214
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### Explicit basis/weight vectors for irreducibles inside the plethysm $Sym^m(\bigwedge^p \mathbf(V))$

This is a follow up to this question about finding the multiplicities of irreducible representations restricted to Young diagrams of 2-columns or less, inside the plethysm $Sym^m(\bigwedge^p \mathbf(V)...

8
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2
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### In general, are 'Young symmetrisers' given by Littlewood-Richardson 'Orthogonal projection Operators'?

Consider $V^{\otimes n}$ where $V$ is vector space and the representation of GL(V) acting in the usual way. Now if I consider tensor products or plethysms of irreducible spaces, this is not in general ...

2
votes

1
answer

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### Does Schur's Lemma hold in this case? Regular representations of $S_n$ over $\mathbb R$

Warning I am a physicist and I am not familiar with a lot of the machinary of representation theory.
I consider the regular representation of $\mathbb S_n$ over reals $\mathbb R$ ($\mathbb R \mathbb ...

1
vote

1
answer

222
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### Homomorphisms from irreducible spaces to reducible spaces

Let $P_{\lambda}$ be a Young symmetriser associated to the following tableau $(a_1 a_2 a_3 b_3 ; b_1 b_2)$ where the entries seperated by the ; belong to first and second COLUMNS of the tableau. Take $...

2
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3
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### 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 ...

8
votes

1
answer

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### 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
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3
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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?
I ...

4
votes

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 ...

11
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1
answer

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### 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 ...

5
votes

1
answer

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### 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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1
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654
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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. ...

6
votes

1
answer

271
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### 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
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1
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### 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: $Sym^2(Sym^k(\mathbb{C}^n))$, $Sym^2(\wedge^k\...

5
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3
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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 ...

9
votes

2
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### 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 ...

31
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6
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### What is known about this plethysm?

Let $S^{\lambda}$ be a Schur functor. Is there a known positive rule to compute the decomposition of $S^{\lambda}(\bigwedge^2 \mathbb{C}^n)$ into $GL_n(\mathbb{C})$ irreps?
In response to Vladimir's ...