On certain graded Sn-modules and the q-Kostka polynomials

Let Λ be the space of symmetric functions and V k be the subspace spanned by the modified Schur functions {S λ [X/(1 − t)]} λ1≤k . We introduce a new family of symmetric polynomials, {A, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials A (k) λ [X; t] form a basis for V k and that the Macdonald polynomials indexed by partitions whose first part is not larger than k expand positively in terms of our polynomials. A proof of this conjecture would not only imply the Macdonald positivity conjecture, but would substantially refine it. Our construction of the A (k) λ [X; t] relies on the use of tableaux combinatorics and yields various properties and conjectures on the nature of these polynomials. Another important development following from our investigation is that the A (k) λ [X; t] seem to play the same role for V k as the Schur functions do for Λ. In particular, this has led us to the discovery of many generalizations of properties held by the Schur functions, such as Pieri and Littlewood-Richardson type coefficients.

show abstract

“…, ρ m ). For example, w = 131332 has degree 6 and evaluation (2,1,3). A word w of degree n is standard iff its evaluation is (1, .…”

Section: Tableaux Combinatoricsmentioning

confidence: 99%

Tableau atoms and a new Macdonald positivity conjecture

Lapointe¹,

Lascoux²,

Morse³

2003

Duke Math. J.

132

View full text Add to dashboard Cite

show abstract

“…The case of sl 2 was proven in [6] by proving 2.13 (in this case, the multiplicities are the usual co-charge Kostka polynomials). Conjecture 2.11 was proven for sl n symmetric-power representations in [15] by using a result of [9]. In [2], we proved 2.13 for sl n KR-modules by using to a result of [18] for the fermionic form of generalized Kostka polynomials, which are the q-multiplicities in the case of tensor products of KR-modules of sl n .…”

Section: Q-systems Kirillov and Reshetikhinmentioning

confidence: 80%

Proof of the Combinatorial Kirillov-Reshetikhin Conjecture

Francesco

Kedem

2008

International Mathematics Research Notices

View full text Add to dashboard Cite

Abstract. In this paper we give a direct proof of the equality of certain generating function associated with tensor product multiplicities of Kirillov-Reshetikhin modules of the untwisted Yangian for each simple Lie algebra g. Together with the theorems of Nakajima and Hernandez, this gives the proof of the combinatorial version of the Kirillov-Reshetikhin conjecture, which gives tensor product multiplicities in terms of restricted fermionic summations.

show abstract

“…Garsia and Procesi [5] gave a purely algebraic construction of the graded S n -module H • (B μ ) in terms of quotients of the coinvariant algebra of symmetric groups as well as a proof of Theorem 2.2.…”

Section: Basics On Kostka Polynomialsmentioning

confidence: 99%