Bases for diagonally alternating harmonic polynomials of low degree

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:118

Article

Bergeron, Nantel¹  Chen, Zhi¹ 

[1] York Univ, Dept Math & Stat, Toronto, ON M3J 1P3, Canada

关键词: Alternating;    Diagonal harmonic;    Explicit basis;    Low degree;    Jeu de Taquin;    Schensted insertion;    Tableaux;   

DOI  :  10.1016/j.jcta.2010.04.002

来源: Elsevier

【 摘 要 】

Given a list of n cells L = [(P(1), q(1)), (p(n), c(n))] where p(1) q(1) is an element of Z >= 0 we let Delta(L) = det parallel to(p(j)(1))(-1)(q(j)(1))(-1) x(i)(pj) y(i)(qj)parallel to The space of diagonally alternating polynomials is spanned by [Delta L] where L varies among all lists with n cells For a > 0 the operators E(a) = Sigma(n)(i=1) y(i)partial derivative(a)(xi) act on diagonally alternating polynomials Haiman has shown that the space An of diagonally alternating harmonic polynomials is spanned by {E(lambda)Delta(n)} where lambda = (lambda(1) lambda(l)) varies among all partitions E(lambda) = E(lambda 1) E(lambda l) and Delta(n) = det parallel to((n -j)(1))(-1)x(i)(n-j)parallel to For t = (t(m), t(1)) is an element of Z(>0)(m) with t(m) > > t(1) > 0 we consider here the operator F(t) = det parallel to E(tm-j+1)+(j-1)parallel to Our first result is to show that F(t)Delta(L) is a linear combination of Delta(L) where L' is obtained by moving l(t)=m distinct cells of L in some determined fashion This allows us to control the leading term of some elements of the form F(t(1)) F(t(r))Delta(n) We use this to describe explicit bases of some of the bihomogeneous components of A(n) = circle plus A(n)(kl) where A(n)(kl) = Span{E(lambda)Delta(n) l(lambda) = l, vertical bar lambda vertical bar = k} More precisely we give an explicit basis of A(n)(kl) whenever k < n To this end we introduce a new variation of Schensted insertion on a special class of tableaux This produces a bijection between partitions and this new class of tableaux The combinatorics of these tableaux T allow us to know exactly the leading term of F(T)Delta(n) where F(T) is the operator corresponding to the columns of T whenever n is greater than the weight of T (C) 2010 Elsevier Inc All rights reserved

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