A Littlewood-Richardson rule for factorial Schur functions

Abstract

Abstract. We give a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial Schur functions. It is a special case of a more general rule which also gives the coefficients in the expansion of a skew factorial Schur function. Applications to Capelli operators and quantum immanants are also given.

On reflection algebras and twisted Yangians

“…It is instructive for what follows to recall in more detail the basic definitions associated to Yangians [8,27] (for a review on Yangians see e.g. [28]). The gl(n) Yangian Y, is a non abelian algebra -a quantum group [9]-[10]-with generators Q (p) ab and defining relations given below…”

Untitled

“…We note that we obtain a generalization of the full equivariant Pieri formula, and not only the Chevalley formula for divisors (see, for example, [Knutson and Tao 2003;Mihalcea 2006;Lakshmibai et al 2006;Kostant and Kumar 1986;Molev and Sagan 1999;Okun'kov and Ol'shanskiȋ 1997] for various forms of the latter formula). This general form was first given by T. Santiago and we essentially reproduce the calculations of [Santiago 2006] in our language.…”

A Chevalley formula for the equivariant quantum $K$-theory of cominuscule varieties

“…It was shown in [Mih07] that the same method can be applied to arbitrary flag varieties. Molev-Sagan-type equations have by now been used to prove Littlewood-Richardson rules in several papers [MS99,KT03,Buc15,PY17].…”

“…It is known that the Molev-Sagan equations together with known expressions [KK86] for the special coefficients C w w,w uniquely determine these structure constants [MS99,KT03,Mih07]. It is interesting to note that every solution {D w u,v } to the Molev-Sagan equations must satisfy the commutativity relation D w u,v = D w v,u , but we will not use this fact.…”

“…The following instances of the additional condition of Corollary 5.5 have appeared: D w w,w = C w w,w in [MS99,KT03], D w w,1 = 1 in [Mih07], and D w 1,1 = δ 1,w in [Mih07].…”