For each pair of positive integers (k, r) such that k + 1, r − 1 are coprime, we introduce an ideal I (k,r) n of the ring of symmetric polynomials C[x1, · · · , xn] Sn. (k,r) n has a basis consisting… (More)

We consider the XXX-type and Gaudin quantum integrable models associated with the Lie algebra gl N. The models are defined on a tensor product M 1 ⊗. .. ⊗ M n of irreducible gl N-modules. For each… (More)

For each pair (k, r) of positive integers with r ≥ 2, we consider an ideal I (k,r) n of the ring of symmetric polynomials. The ideal I (k,r) n has a basis consisting of Macdonald polynomials P λ (x1,… (More)

We define the Hopf algebra structure on the Grothendieck group of finitedimensional polynomial representations of Uq ĝlN in the limit N → ∞. The resulting Hopf algebra Rep Uq ĝl∞ is a tensor product… (More)

We prove the B. and M. Shapiro conjecture that says that if the Wronskian of a set of polynomials has real roots only, then the complex span of this set of polynomials has a basis consisting of… (More)

We consider critical points of master functions associated with integral dominant weights of Kac-Moody algebras and introduce a generating procedure constructing new critical points starting from a… (More)

We show that the Gaudin Hamiltonians H 1 ,. .. , H n generate the Bethe algebra of the n-fold tensor power of the vector representation of gl N. Surprisingly the formula for the generators of the… (More)

We prove that if all roots of the discrete Wronskian with step 1 of a set of quasi-exponentials with real bases are real, simple and differ by at least 1, then the complex span of this set of… (More)

We show that the Bethe vectors are non-zero vectors in the sl r+1 Gaudin model. Namely, we show that the norm of a Bethe vector is equal to the Hessian of the corresponding master function at the… (More)

Bosonic formulas for generating series of partitions with certain restrictions are obtained by solving a set of linear matrix q-difference equations. Some particular cases are related to… (More)