# Dimitry Gurevich

## Reflection Equation Algebra and related combinatorics

Reflection Equation Algebras constitute a subclass of the so-called Quantum Matrix algebras. Each of the REA is associated with a quantum $R$-matrix. In a sense the REA corresponding to quantum $R$-matrix of Hecke type can be considered as $q$-counterparts of the commutative algebra $Sym(gl(N))$ or the enveloping algebras $U(gl(N))$. On any such RE algebra there exist analogs of some symmetric polynomials, namely the power sums and the Schur functions. In my talk I plan to exhibit $q$-versions of the Capelli formula, the Frobenius formula, related to this combinatorics. Also I plan to introduce analogs of the Casimir operators and partially perform their spectral analysis.