Young Tableaux, Canonical Bases, and the Gindikin-Karpelevich Formula

Abstract

Abstract. A combinatorial description of the crystal B(∞) for finitedimensional simple Lie algebras in terms of certain Young tableaux was developed by J. Hong and H. Lee. We establish an explicit bijection between these Young tableaux and canonical bases indexed by Lusztig's parametrization, and obtain a combinatorial rule for expressing the Gindikin-Karpelevich formula as a sum over the set of Young tableaux. IntroductionThe Gindikin-Karpelevich formula is a p-adic integration formula proved by Langlands in [18]. He named it the Gindikin-Karpelevich formula after a similar formula originally stated by Gindikin and Karpelevich [5] in the case of real reductive groups. The formula also appears in Macdonald's work [25] on p-adic groups and affine Hecke algebras.Let G be a split semisimple algebraic group over a p-adic field F with ring of integers o F , and suppose the residue field o F /πo F of F has size t, where π is a generator of the unique maximal ideal in o F . Choose a maximal torus T of G contained in a Borel subgroup B with unipotent radical N , and let N − be the opposite group to N . We have B = T N . The group G(F ) has a decomposition G(F ) = B(F )K, whereis the modular character of B and τ is extended to B(F ) to be trivial on N (F ). The function f• is called the standard spherical vector corresponding to τ .Let G ∨ be the Langlands dual of G with the dual torus T ∨ . The set of coroots of G is identified with the set of roots of G ∨ and will be denoted by Φ. Finally, let z be the element of the dual torus T ∨ , corresponding to τ via the Satake isomorphism.Theorem 0.1 (Gindikin-Karpelevich formula, [18]). Given the setting above, we havewhere Φ + is the set of positive roots of G ∨ .Let g be the Lie algebra of G ∨ , and let B(∞) be the crystal basis of the negative part U − q (g) of the quantum group U q (g). Then Φ is the root system of g as well. In recent work, the integral in the Gindikin-Karpelevich formula has been evaluated using Kashiwara's crystal basis or Lusztig's canonical basis. D. Bump and M. Nakasuji where nz(φ i (b)) is the number of nonzero entries in the Lusztig parametrization φ i (b) of b with respect to a reduced expression i of the longest Weyl group element. The purpose of this paper is to describe the sum in (0.2) in a combinatorial way using Young tableaux. Since the canonical basis B is the same as Kashiwara's global crystal basis, we may replace B with B(∞). The associated crystal structure on B may be described in terms of the Lusztig parametrization [2,24] or the string parametrization [1,13,22], and there are formulas relating the two given by Berenstein and Zelevinsky in [2]. Much work has been done on realizations of crystals (e.g., [8,9,15,16,21]). In the case of B(∞) for finitedimensional simple Lie algebras, J. Hong and H. Lee used marginally large semistandard Young tableaux to obtain a realization of crystals [7]. We will use their marginally large semistandard Young tableaux realization of B(∞) to write the right-hand side of (0.2) as a sum over a set T (∞) ...

Crystal ℬ(∞) for Quantum Affine Algebras: A Young Wall Description

“…Such results were previously obtained in [18,19] for the finite simple Lie algebra types, using the results of [4], and the marginally large walls introduced in the current work should allow an analogous treatment of the affine types.…”

Atomic decomposition of characters and crystals

“…. We refer to [22] for a complete description of the bijection Ξ in each case. In the sequel, we only need the following two properties of the map Ξ.…”

The flush statistic on semistandard Young tableaux

“…(1) Let T ∈ B(λ + ρ) be a tableaux. We define a k-segment [15,16] of T (in the ith row) to be a maximal consecutive sequence of k-boxes in the ith row for any i + 1 ≤ k ≤ r + 1. Denote the total number of k-segments in T by seg(T ).…”

“…This formula, from the context of crystals, may be viewed as the Verma module analogue of the highest weight calculation used in the Casselman-Shalika formula. In [15,16], we were able to recover the path to the highest weight vector using the marginally large tableaux of J. Hong and H. Lee [10], which is a certain enlargement of semistandard Young tableaux, and interpret the decorations. The corresponding statistic was called the segment statistic and may be easily read off from a marginally large tableaux.…”

“…The corresponding statistic was called the segment statistic and may be easily read off from a marginally large tableaux. Outside of type A r , the proof in [15] relied on the interpretation of the Gindikin-Karpelevich formula as a sum over Lusztig's canonical basis given by H. H. Kim and K.-H. Lee in [12,13] and did not require any decorated paths to the highest weight in the crystal graph.…”