Questions tagged [multisymmetric-functions]
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5 questions
8 votes
1 answer
361 views
Commuting variety and invariant polynomials
Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{sl}_n\mathbb{C}$, and let $S_n$ denote the Weyl group of $(\mathfrak{sl}_n\mathbb{C}, \mathfrak h)$. I am interested in understanding the ...
- 2,373
3 votes
1 answer
624 views
A generalization of Newton-Girard Identities
Let $x_1, ..., x_n$ be formal variables. One variant of the Newton-Girard identities expresses $$\sum_{\pi \in S_n} x_{\pi(1)} x_{\pi(2)} \cdots x_{\pi(k)}$$ as a polynomial in the power sums of the $...
- 15.7k
2 votes
1 answer
149 views
What $n$-linear sums can be extracted from a product of linear polynomials in $m$ variables?
Let $\boldsymbol{c}_1, ..., \boldsymbol{c}_n$ be $n$ orthonormal, $m$-dimensional complex vectors, with $\boldsymbol{c}_i = (c_{i,1}, ..., c_{i,m})$. Consider the following polynomial in $x_1,..., x_m$...
- 352
8 votes
1 answer
1k views
Grassmann–Plücker relations for permanents
Let $K$ be a field, $1 \leq d \leq n$ integers and $V$ an $n$-dimensional vector space. The Grassmann–Plücker relations are quadratic forms on $\bigwedge^d V$ whose zero set is exactly the set of ...
- 25.2k
15 votes
3 answers
3k views
which homogeneous polynomials split into linear factors?
Let $R$ be the set of homogeneous polynomials of degree $n$ in $d$ variables over $\mathbb{C}$. When $n>2$, the set of elements of $R$ that split into a product of linear factors forms a proper ...
- 323