TY - JOUR

T1 - Skew Howe duality and q-Krawtchouk polynomial ensemble

AU - Nazarov, Anton

AU - Nikitin, Pavel

AU - Sarafannikov, Daniil

N1 - 17 pages, 5 figures, submitted to Zapiski Nauchnykh Seminarov POMI

PY - 2022/12/22

Y1 - 2022/12/22

N2 - We consider the decomposition into irreducible components of the exterior algebra $\bigwedge\left(\mathbb{C}^{n}\otimes \left(\mathbb{C}^{k}\right)^{*}\right)$ regarded as a $GL_{n}\times GL_{k}$ module. Irreducible $GL_{n}\times GL_{k}$ representations are parameterized by pairs of Young diagrams $(\lambda,\bar{\lambda}')$, where $\bar{\lambda}'$ is the complement conjugate diagram to $\lambda$ inside the $n\times k$ rectangle. We set the probability of a diagram as a normalized specialization of the character for the corresponding irreducible component. For the principal specialization we get the probability that is equal to the ratio of the $q$-dimension for the irreducible component over the $q$-dimension of the exterior algebra. We demonstrate that this probability distribution can be described by the q-Krawtchouk polynomial ensemble. We derive the limit shape and prove the central limit theorem for the fluctuations in the limit when $n,k$ tend to infinity and $q$ tends to one at comparable rates.

AB - We consider the decomposition into irreducible components of the exterior algebra $\bigwedge\left(\mathbb{C}^{n}\otimes \left(\mathbb{C}^{k}\right)^{*}\right)$ regarded as a $GL_{n}\times GL_{k}$ module. Irreducible $GL_{n}\times GL_{k}$ representations are parameterized by pairs of Young diagrams $(\lambda,\bar{\lambda}')$, where $\bar{\lambda}'$ is the complement conjugate diagram to $\lambda$ inside the $n\times k$ rectangle. We set the probability of a diagram as a normalized specialization of the character for the corresponding irreducible component. For the principal specialization we get the probability that is equal to the ratio of the $q$-dimension for the irreducible component over the $q$-dimension of the exterior algebra. We demonstrate that this probability distribution can be described by the q-Krawtchouk polynomial ensemble. We derive the limit shape and prove the central limit theorem for the fluctuations in the limit when $n,k$ tend to infinity and $q$ tends to one at comparable rates.

KW - math.RT

KW - math-ph

KW - math.MP

KW - math.PR

KW - 22E46, 33C45, 05E05, 60G55, 60B10, 17B10

KW - предельная форма

KW - диаграмма Юнга, q-полиномы Кравчука

KW - детерминантный ансамбль

KW - q-размерность

KW - ортогональные полиномы

UR - https://www.mathnet.ru/rus/znsl7283

M3 - Article

VL - 517

SP - 106

EP - 124

JO - ЗАПИСКИ НАУЧНЫХ СЕМИНАРОВ САНКТ-ПЕТЕРБУРГСКОГО ОТДЕЛЕНИЯ МАТЕМАТИЧЕСКОГО ИНСТИТУТА ИМ. В.А. СТЕКЛОВА РАН

JF - ЗАПИСКИ НАУЧНЫХ СЕМИНАРОВ САНКТ-ПЕТЕРБУРГСКОГО ОТДЕЛЕНИЯ МАТЕМАТИЧЕСКОГО ИНСТИТУТА ИМ. В.А. СТЕКЛОВА РАН

SN - 0373-2703

ER -