Abstract
We prove that the Lam-Shimozono ``down operator'' on the affine Weyl group induces a derivation of the affine Fomin-Stanley subalgebra of the affine nilCoxeter algebra. We use this to verify a conjecture of Berg, Bergeron, Pon and Zabrocki describing the expansion of k-Schur functions of ``near rectangles'' in the affine nilCoxeter algebra. Consequently, we obtain a combinatorial interpretation of the corresponding k-Littlewood–Richardson coefficients. Nous montrons que l’opérateur ``down'', défini par Lam et Shimozono sur le groupe de Weyl affine, induit une dérivation de la sous-algèbre affine de Fomin-Stanley de l'algèbre affine de nilCoxeter. Nous employons cette dérivation pour vérifier une conjecture de Berg, Bergeron, Pon et Zabrocki sur l'expansion des k-fonctions de Schur indexées par les partitions qui sont ``presque rectangles''. Par conséquent, nous obtenons une interprétation combinatoire des k-coefficients de Littlewood–Richardson correspondants.
Highlights
1 Introduction k-Schur functions were first introduced by Lapointe, Lascoux and Morse [13] in the study of Macdonald polynomials
This was done by identifying the algebra of k-Schur functions with the affine Fomin-Stanley subalgebra of the affine nilCoxeter algebra A [8]
Lam [7] pointed out that the k-Littlewood–Richardson coefficients are the same coefficients that appear in the expansion of a k-Schur function in the standard basis of A
Summary
We let W denote the affine symmetric group with generators si for i ∈ I, and relations s2i = 1, sisj = sjsi, when i and j are not adjacent, and sisjsi = sjsisj when i and j are adjacent. An element of the affine symmetric group may be expressed as a word in the generators si. An element of the affine symmetric group may have multiple reduced words, words of minimal length which express that element. The length of w, denoted (w), is the number of generators in any reduced word of w. The Bruhat order on affine symmetric group elements is a partial order where v < w if there is a reduced word for v that is a subword of a reduced word for w. We denote by W j the set of minimal length representatives of the cosets W/Wj
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