Let $A=(a_{ij})$ be an $n\times n$ real matrix and let $\lambda$ be a partition of $n$. Let $\phi^\lambda$ be the class function dual to the Young permutation character, and let $$ \phi^\lambda[A] = \sum_{\sigma\in\mathfrak{S}_n}\phi^\lambda(\sigma) \prod_{i=1}^{n}a_{i\sigma(i)} $$ be the corresponding monomial immanant. Stembridge [Canad. J. Math. 44 (1992), pp. 1079-1099] posed the following open problem: If all minors of $A$ of order at most $r$ are nonnegative, and the partition $\lambda$ has length at most $r$, is it true that $\phi^\lambda[A]\ge 0$? This paper solves the problem for all hook partitions.
Let $\mathfrak{g}$ be a complex semisimple Lie algebra and, for a dominant integral weight $\lambda$, let $\Theta_\lambda$ be the multiplicity-free sum of the weights of the irreducible module $V(\lambda)$. The $\Theta_\mu$ form a $\mathbb{Z}$-basis of the $W$-invariants, so there are unique integers $a(\mu,\lambda)$,...
Felipe de Mattos Chafik Hindi, W. Schutzer· 0 citations
Let $\mathcal{S}^ r$ denote the Schatten--von Neumann class and let $S_{\Psi_{f,\lambda}}$ be the Schur--Hadamard multiplier whose symbol is the divided-difference matrix of $f$ along $\lambda$. Let $0<\alpha,r<\infty$, let $f\in C^1([-1,1])$ satisfy $f(0)=0$, $|f'(t)|\lesssim |t|^\alpha$, and let $\lambda\in\ell^r$ be...
. Let h n denote the complete homogeneous symmetric function of degree n , s ν the Schur function indexed by a partition ν , and ⟨· , ·⟩ the Hall inner product. The plethysm coefficient ⟨ h n [ h m ] ,s ν ⟩ is the multiplicity of the Weyl module W ν ( C N ) in Sym n (Sym m ( C N )) whenever ν has at most N parts. For a...
J. Farkas, Károly Jármai· Analysis and Optimum Design...· 0 citations
Let $n>k$ be positive integers and let $\varepsilon,\delta\in\{-1,1\}$. In this note, we classify the unimodular zeros of the harmonic trinomial $H_{\varepsilon,\delta}:\mathbb{C}\to \mathbb{C}$ given by \[ H_{\varepsilon,\delta}(z)=z^n+\varepsilon\overline z^{\,k}+\delta, \] where $\overline{z}$ is the complex conjuga...
Let $\Phi$ be an irreducible root system normalized so that long roots have length $\sqrt{2}$, and let $\mu\leq\lambda$ be dominant weights. We associate to $(\Phi,\mu,\lambda)$ the convex polytope $$P^{\Phi}_{\mu,\lambda} = (\mu+C)\cap(\lambda-C)\cap(\overline{D}-\rho),$$ where $C=\mathbb{R}_{\geq 0}\Phi^+$ and $\over...
Let $K$ be a number field and $S$ a finite set of non-archimedean places. Write $\mathcal{O}_S$ for the ring of $S$-integers of $K$ and $\mathcal{O}_S^\times$ for its unit group. Let $\pi : X \rightarrow \mathbb{P}^1$ be a morphism of (irreducible) curves defined over $K$, and denote by $\operatorname{Red}(\pi)$ the se...
Michael Stoll, S. Siksek· 0 citations
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