Questions tagged [schur-functions]
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Are the minors of this Hadamard product Schur positive?
Let $h_i (x)$ denote the complete symmetric function of degree $i$ in some set of variables $x = (x_1 , x_2 , \dots)$. Then the minors of the Toeplitz matrix $T (x) = \left(h_{i-j} (x) \right)_{i,j}$ ...
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Applying $\sum_i \partial_{x_i}$, $\sum_i x_i \partial_{x_i}$ and $\sum_i x_i^2 \partial_{x_i}$ to Schur polynomials
The operators $L_k=\sum_i x_i^k\frac{\partial}{\partial x_i}$, with integer $k$, take symmetric polynomials into symmetric polynomials.
Is it known how to write the result of the application of $L_0$, ...
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Factorization of symmetric polynomials
Let $\Lambda_n$ be the algebra of all symmetric polynomials in $n$ variables, which we also consider as an infinite-dimensional vector $\mathbb{Q}$-space, whose basis is the Schur polynomials.
The ...
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Significance of partition containment in representation theory of $\operatorname{GL}_n$
I'm working on a Schur positivity problem and I came across a series of Schur polynomials (all in $n$ variables) whose indexing partitions are (conjecturally) contained in one of some given set of ...
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Is the appearance of Schur functions a coincidence?
The Schur functions are symmetric functions which appear in several different contexts:
The characters of the irreducible representations for the symmetric group (under the characteristic isometry).
...
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"Sufficiently fat" Schur polynomial times Schubert polynomial is Schubert-positive
It's still open to find a combinatorial proof that for integers $k>0$ and $m>0$, a permutation $u\in S_\infty$ with its final descent at position $m$, and a partition $\lambda$ that
$$\mathfrak{...
- 2,168
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Representation-theoretic interpretation of double Schur polynomials
The Schur polynomials
$$s_\lambda(x_1, \ldots, x_n) = \frac{|x_i^{\lambda_j+n-j}|_{1\le i,j\le n}}{|x_i^{n-j}|_{1\le i,j\le n}}$$
naturally appear as polynomial representatives for Schubert classes in ...
- 3,084
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Is this simple symmetry of Littlewood-Richardson coefficients known?
Let $\lambda$ be a partition with at most $p$ parts, let $\mu$ be a partition with at most $q$ parts, and let $\nu$ be a partition with at most $p+q$ parts. Let $m\geq \nu_1$ be an integer. We denote ...
- 2,168
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The convex hull of Schur polynomial evaluations
Let $r\leq n$ and $d$ be positive integers. A probability vector is a vector of non-negative entries that sum to 1. For each probability vector $\lambda$ of length $n$, let
$$s(\lambda)=(\dim[\pi] \...
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About Cauchy identity for Schur polynomials
(This was originally posted here, https://math.stackexchange.com/questions/4687466/cauchy-identity-for-schur-functions, and I am reposting it here as it seems to be more appropriate.)
PRELIMINARY.
The ...
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Dose density matrix with off-diagonal elements equal to zero has maximum von-Neumann entropy?
von-Neumann entropy
I know von-Neumann entropy on density matrix $S=-{\rm Tr}(\rho \ln\rho)$ is similar to Shannon entropy $S=-\sum_i p_i\ln p_i$ in classical mechanics. And I want to get Bose-...
- 3
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Sum of Schur functions associated to self-conjugate partitions
The $\tau$-function $H^\circ \big(t ;\vec{x} \big)$ associated with counting simple Hurwitz numbers is the formal power series
\begin{equation}
(\dagger) \quad H^\circ \big(t ;\vec{x} \big) \, =
\,
\...
- 1,847
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Identities involving Littlewood–Richardson coefficients?
I am not aware of that many identities that involve several Littlewood–Richardson coefficients.
One recent identity, is a generating function as sum of squares of LR-coefficients,
due to Harris and ...
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Harmonic flow on the Young lattice
Let me begin with some preliminary concepts: A positive real-valued function $\varphi: P \rightarrow \Bbb{R}_{>0}$ on a locally finite, ranked poset $(P, \trianglelefteq)$ is harmonic if
$\varphi(\...
- 1,847
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Nonnegativity locus of Schur polynomials
Let $a_1,\ldots,a_n \in \mathbb{C}$ be complex numbers that are the zeros of a real polynomial (meaning that the non-real ones come in complex conjugate pairs). Suppose that these numbers are such ...
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