Algebraic Combinatorics Seminar

Title: Asymptotics of the principal of specializations of Schubert polynomials

Abstract: Schubert polynomials were introduced by Lascoux and Sch\"utzenberger in 1982 to study Schubert varieties. They have been intensely studied since and remain a central object in algebraic combinatorics. Macdonald showed in 1991 that the principal specialization, i.e. setting all variables to one, gives a weighted sum over reduced words. In 2017 Stanley conjectured that the limit of this specialization exists as n goes to infinity and asked the question of what kind of permutations maximize the value for fixed n. Merzon and Smirnov conjectured that this maximum is achieved on layered permutations. We resolve Stanley’s problem restricted to layered permutations and find the shape of such a permutation. Joint work with Igor Pak and Greta Panova. No knowledge of Schubert polynomials is required.