**11**

votes

**4**answers

3k views

### Expressing power sum symmetric polynomials in terms of lower degree power sums

Is there an explicit formula expressing the power sum symmetric polynomials
$$p_k(x_1,\ldots,x_N)=\sum\nolimits_{i=1}^N x_i^k = x_1^k+\cdots+x_N^k$$
of degree $k$ in $N < k$ variables entirely ...

**8**

votes

**3**answers

648 views

### Arithmetic product of symmetric functions: why is it integral?

For every commutative ring $A$, let $\mathbf{Symm}_A$ be the ring of symmetric functions over $A$. Let $\mathbf{Symm}$ without a subscript denote $\mathbf{Symm}_{\mathbb{Z}}$.
We can define a ...

**20**

votes

**1**answer

547 views

### Bounding Schur symmetric polynomials on the unit circle

Recall the Schur polynomial in $n$ variables, indexed by the partition $\lambda$, with $\ell(\lambda) \leq n$, is given by
\begin{equation}
s_\lambda(x_1,\ldots, x_n) = a_{\lambda + \delta}(x_1, \...

**15**

votes

**1**answer

285 views

### Cohomology of configuration space as a representation of the symmetric group

Let $X_n$ be the space of $n$ distinct labeled points in $\mathbb{R}^3$, which is equipped with an action of the symmetric group $S_n$. It is well known that the total cohomology of $X_n$ is ...

**4**

votes

**2**answers

415 views

### Cyclically symmetric functions

Where can I learn about the invariant theory associated with actions of cyclic groups (as opposed to symmetric groups)?
E.g., do the functions $x+y+z$, $xy+yz+zx$, and $x^2y+y^2z+z^2x$ generate the ...

**2**

votes

**1**answer

62 views

### Reference or a short argument that a certain subset generates the ring of p-typical symmetric functions under plethysm

Let p be a prime, and let $\Lambda_p$ be the subring of the ring of symmetric functions $\Lambda$ (over $\mathbb{Z}$) such that $$x \in \Lambda_p$$ iff there is an $i \in \mathbb{N}$ such that $p^ix \...

**9**

votes

**1**answer

344 views

### Restriction of characters of hyperoctahedral groups.

The hyperoctahedral group $H_n$ has several descriptions; as a wreath product; as signed permutation matrices; as the Weyl group of type $B_n$ or $C_n$. In all these descriptions it is apparent that ...

**6**

votes

**2**answers

698 views

### Expression for the sum of square roots of zeros of a polynomial

Let $f(x)$ be a polynomial of degree $n$ with rational coefficients whose zeroes are nonnegative real numbers: $x_1, \dots, x_n\geq 0$.
General question. Does there exist a simple expression for the ...

**7**

votes

**0**answers

219 views

### Deformation of the product of symmetric functions

Matthias Lederer and I are studying a deformation of the Littlewood-Richardson product of Schur functions. It's a bit complicated to define (and work in progress) so I won't give the full definition ...

**5**

votes

**1**answer

242 views

### Is the “renormalized third comultiplication” on $\mathbf{Symm}$ integral?

Background:
For any commutative ring $R$, let $\mathbf{Symm}_R$ be the ring of symmetric functions in countably many variables $x_1$, $x_2$, $x_3$, ... over $R$. ("Symmetric functions" really means ...

**4**

votes

**0**answers

165 views

### Anti-arithmetic product of symmetric functions: (why) is it integral?

This is an analogue of MathOverflow question #138148. Indeed it is so analogous that I wrote the following by copypasting said question and making the necessary changes.
For every commutative ring $A$...

**3**

votes

**0**answers

122 views

### A “nice” Orthogonal Basis for Translation Invariant Symmetric Polynomials

It is going to be a rather long question, so I will first state it and then try to explain and motivate it.
Take $\Lambda_n $ as the graded ring of symmetric polynomials of a field $F$ in $n$ ...

**3**

votes

**0**answers

125 views

### 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**answers

176 views

### 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 ...