Kazhdan-lusztig Polynomials and Canonical Basis

In this paper we show that the Kazhdan–Lusztig polynomials (and, more generally, parabolic KL polynomials) for the group Sn coincide with the coefficients of the canonical basis in nth tensor power of the fundamental representation of the quantum group Uqslk. We also use known results about canonical bases for Uqsl2 to get a new proof of recurrent formulas for KL polynomials for maximal parabolic subgroups (geometrically, this case corresponds to Grassmanians), due to LascouxSchützenberger and Zelevinsky. 1. Review of the theory of Kazhdan-Lusztig polynomials In this section, we review the theory of Kazhdan-Lusztig polynomials. We will use their generalization to the parabolic case, defined by Deodhar (see [D]). For the sake of completeness and to fix notations, we list the main definitions and results here, referring the reader to the original papers for more details. To avoid confusion with the theory of quantum groups, we will not use the variable q in the definition of the Hecke algebra; instead, we will use v = q. Let W be a finite Weyl group with a set of simple reflections S. We denote by l(w) the length of w ∈W with respect to the generators s ∈ S. Let H(W ) be the Hecke algebra associated with W (we will usually omitW and write just H). By definition, it is an associative algebra with unit over the field Q(v) with generators Ts, s ∈ S and relations (1.1) TsTs′ · · · = Ts′Ts . . . n terms on each side where n is the order of ss in W (Ts + 1)(Ts − v ) = 0. For any subset J ⊂ S, let WJ be the parabolic subgroup in W generated by s ∈ J . Denote by W J the set of minimal length representatives of the cosets in W/WJ , and let < be the partial order on W J induced by the Bruhat order on W . Let M be a vector space over Q(v) with the basis my , y ∈W J . Typeset by AMS-TEX 1 Proposition 1.1(see [D, Lemmas 2.1, 2.2]). Let u be either −1 or v. (1) The following formulas define the structure of an H-module on M : (1.2) Tsmσ =    vmsσ + (v −2 − 1)mσ, l(sσ) < l(σ), vmsσ l(sσ) > l(σ), sσ ∈W J umσ, l(sσ) > l(σ), sσ / ∈W J Note that if σ ∈W J , l(sσ) < l(σ) then sσ ∈W J . (2) If σ ∈W J then Tσme = v mσ. Remarks. 1. Our notations slightly differ from those of Deodhar: what we denote mσ in his notations would be q mσ. Note also that there is a misprint in the formula for L(s) (immediately after the statement of Lemma 2.1) in [D]. 2. It is easy to see thatM is a deformation of the induced representation IndWWJ1 of W . In particular, if J = ∅ then M is the left regular representation of H and the action is independent of u. Define an involution : Q(v) → Q(v) by f(v) 7→ f(v). We will say that a map φ of Q(v)-vector spaces is antilinear if φ(fx) = f̄φ(x) for any f ∈ Q(v) and any vector x. We define antilinear involutions on H and M by letting Ts = T −1 s = v Ts+ v 2 − 1, me = me and ab = āb̄. Note that one hasmσ = (vTσ)me = mσ+ ∑ τ<σ cτσmτ . Theorem 1.2([D, Proposition 3.2]). Let us assume that we have fixed u as in Proposition 1.1. Then for every σ ∈ W J there exists a unique element Cσ ∈ M such that the following two conditions are satisfied: (1.3) Cσ =Cσ, Cσ = ∑