Tagged Questions

Symmetric functions are symmetric polynomials, in finitely many, or countably infinitely many variables. They arise in the representation theory of symmetric groups and in the polynomial representation theory of general linear groups. Bases of the ring of symmetric functions are indexed by integer ...

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Symmetric tensors as sum of powers

I am looking for formulas for writing a basis element of $Sym^k(H)$ as sum of elements of the form $v^{\otimes k}$ where $v\in H$. Here $H$ is a hilbert space and by basis element I mean the ...
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Are the Schur functions the minimal basis of the ring of symmetric functions with the following properties?

Let $\Lambda$ denote the ring of symmetric functions in variables $x_1,x_2,\dots$ and with coefficients in $\mathbf{Q}$. Then $\Lambda$ is freely generated as an $\mathbf{Q}$-algebra by $p_1,p_2,\dots$...
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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 ...
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Skew zonal polynomials, skew zonal spherical functions, and combinatorics

Zonal polynomials may be expressed in terms of power sums as $$Z_\lambda=n!\sum_\nu \frac{1}{z_\nu }2^{n-\ell(\nu)}\omega_\lambda(\nu)p_\nu,$$ with usual notation in which $\omega_\lambda(\nu)$ are ...
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sum of squares of Schur polynomials indexed over partition valued functions on a set

Fix a finite set $X$ and two natural numbers $d$ and $n$. For a partition $\lambda$ and a number $d$ denote by $s_\lambda^d(x_1,\dots,x_d)$ the Schur polynomial in $d$-many variables $x_1,\dots,x_d$. ...
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universality of Macdonald polynomials

I have been recently learning a lot about Macdonald polynomials, which have been shown to have probabilistic interpretations, more precisely the eigenfunctions of certain Markov chains on the ...
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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 ...
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Is there a short proof that the Kostka number $K_{\lambda \mu}$ is non-zero whenever $\lambda$ dominates $\mu$?

This is maybe a little basic for MathOverflow, but I'm hoping it will get some interesting answers. Let $\unrhd$ be the dominance order on partitions of $n \in \mathbb{N}$. For partitions $\lambda$ ...
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Constant term identity and the Grassmannian Gr(2,6)

The following conjecture is motivated by two different presentations of the affine cone over Grassmannian $Gr(2,6).$ One as a GIT quotient of $Hom(\mathbb{C}^2, \mathbb{C}^6)//SU(2)$ and the other as ...
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Does the ring generated by the odd power sum symmetric functions have a name?

Let $\Lambda$ be the ring of symmetric functions and recall the power sum symmetric function $p_i = \sum x_1^i + x_2^i + \dots$ generate this ring. Let $\tilde\Lambda$ be the ring generated by the odd ...
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An algebraic strengthening of the Saturation Conjecture

The Saturation Conjecture (proved by Knutson-Tao) asserts that $c_{n\mu,n\nu}^{n\lambda}\neq 0\Rightarrow c_{\mu,\nu}^{\lambda} \neq 0$, where $c$ denotes a Littlewood-Richardson coefficient and $n$ ...
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