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**17**

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271 views

### Deforming a basis of a polynomial ring

The ring $Symm$ of symmetric functions in infinitely many variables is well-known to be a polynomial ring in the elementary symmetric functions, and has a $\mathbb Z$-basis of Schur functions $\{S_\...

**9**

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689 views

### Generalization of Cauchy's identity

Let $ s_{\lambda} $ be the schur function associated to the partition $ \lambda $.
Cauchy's identity (as in Macdonald) states that
$$
\sum_{\lambda} s_{\lambda}(X)s_{\lambda}(Y) = \prod_{i,j}(1-...

**6**

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225 views

### Staircase Schur functions squared

Let $\Delta_n$ be the staircase-shaped partition $(n-1,n-2,\dots,1)$. Are there any non-obvious combinatorial objects that index $s_{\Delta_n}^2$? Here, $s_\lambda$ is the Schur function indexed by ...

**5**

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143 views

### a variational problem related to weighted logarithmic capacity

Consider the following multiple contour integral:
$$ \Phi_\lambda := \oint \ldots \oint \prod_{1 \le j < k \le n} (z_j^{-1} - z_k^{-1}) \prod_{j=1}^n \prod_{k=1}^n (1 - z_j x_k)^{-1} \prod_{j=1}^n ...

**4**

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162 views

### Optimization problem involving Multivariate Normal

I use $\phi(t)$ to describe the standard normal distribution density and $\Phi(t)$ as the normal distribution CDF and would like to prove that for all
$n\geq3$, the function:
$$h(\mu_{1},\ldots,\...

**4**

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225 views

### Analogy between canonical basis of U(n_-) and Schur functors, each under restriction

.1. For any category $\mathcal C$, possibly enriched over schemes, define $Rep({\mathcal C})$ to be the functor category ${\mathcal C} \to {\bf Vec}$ with direct sum inherited from $\bf Vec$. (If $\...

**3**

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164 views

### Generalization of Frobenius formula involving Macdonald polynomials

Given a vector $\vec k=(k_1,k_2,\cdots)$ with $k_i$ are non-negative integers, the Newton polynomial $p_{\vec k}(x)$ is defined as
\begin{equation}
p_{\vec k}(x)=\prod_{j=1}^n p_j^{k_j}(x)~,
\end{...

**2**

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79 views

### Bounding Schur polynomials of a particular shape

Consider Schur polynomials $s_\lambda$ with $\lambda = (2m, m, m, \ldots, m, 0)$ and $\ell(\lambda) = n$ (that is, $\lambda$ has $n$ rows). Here $m \gg n$, which, for the sake of concreteness, let's ...

**2**

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102 views

### 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 ...

**1**

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148 views

### 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)...

**1**

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510 views

### Every antisymmetric (alternating) polynomial is divisible by Vandermonde product

I am looking for a proof of the statement:
Every antisymmetric (alternating) polynomial is divisible by Vandermonde product, yielding a symmetric polynomial in the result.
The statement is really ...