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Avital Frumkin

Publications and source records attributed to Avital Frumkin.

4 recordsLinked to original sources
arXiv2015

Diagonals of real symmetric matrices of given spectra as a measure space

The set of diagonals of real symmetric matrices of given non negative spectrum is endowed with a measure which is obtained by the push forward of the Haar measure of the real orthogonal group.\\ We prove that the Radon Nicodym derivation of this measure with respect to the relative Euclidean measure is approximated by the coefficients of a sequence of zonal sphere polynomials corresponding with the given spectrum. There is a striking similarity between the role of the zonal sphere polynomials in the orthogonal case, and that of the Schur function in the Hermitian case.\\ Following this we obtain a combinatorial approximation for the probability of real symmetric matrix of a given spectrum to appear as the sum of two real symmetric matrices, each of a given spectrum. In addition we obtain a real orthogonal analogue to the Zuber Itzykson Harish Chandra integration formula.

math.RT↗
arXiv2004

Rim Hook Tableaux and Kostant's $η$-Function Coefficients

Using a 0/1 encoding of Young diagrams and its consequences for rim hook tableaux, we prove a reduction formula of Littlewood for arbitrary characters of the symmetric group, evaluated at elements with all cycle lengths divisible by a given integer. As an application, we find explicitly the coefficients in a formula of Kostant for certain powers of the Dedekind $η$-function, avoiding most of the original machinery.

math.CO↗
arXiv2002

Hook Interpolations

The hook components of $V^{\otimes n}$ interpolate between the symmetric power $\sym^n(V)$ and the exterior power $\wedge^n(V)$. When $V$ is the vector space of $k\times m$ matrices over $\bbc$, we decompose the hook components into irreducible $GL_k(\bbc)\times GL_m(\bbc)$-modules. In particular, classical theorems are proved as boundary cases. For the algebra of square matrices over $\bbc$, a bivariate interpolation is presented and studied.

math.RT↗