2023/07/27 by Jing, Naihuan, Liu, Yinlong, Zhang, Jian · 1 citation
#05E10 #17B35 #17B66 #Combinatorics (math.CO) #FOS: Mathematics #Primary: 17B37 Secondary: 20G05 #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2307.14573
The classical Capelli identity is an important determinantal identity of a matrix with noncommutative entries that determines the center of the enveloping algebra of the general linear Lie algebra, and was used by Weyl as a main tool to study irreducible representations in his famous book on classical groups. In 1996 Okounkov found higher Capelli identities involving immanants of the generating matrix of U(gl(n)) which correspond to arbitrary orthogonal idempotent of the symmetric group. It turns out that Williamson also discovered a general Capelli identity of immanants for U(gl(n)) in 1981. In this paper, we use a new method to derive a family of even more general Capelli identities that include the aforementioned Capelli identities as special cases as well as many other Capelli-type identities as corollaries. In particular, we obtain generalized Turnbull's identities for both symmetric and antisymmetric matrices, as well as the generalized Howe-Umeda-Kostant-Sahi identities for antisymmetric matrices which confirm the conjecture of Caracciolo, Sokal, and Sportiello.