# An immanant formulation of the dual canonical basis of the quantum polynomial ring.

1 IntroductionContributing to research concerning the Quantum Yang-Baxter Equation and its applications, Drinfeld [6] and Jimbo [9] introduced quantum analogs of classical enveloping algebras of Lie algebras. These quantized enveloping algebras belong to a family of Hopf algebras now known as quantum groups. Lusztig [13] and Kashiwara [10] advanced the representation theory of quantized enveloping algebras by introducing canonical (or crystal) bases for the algebras, which then led to definitions of dual canonical bases for other quantum groups, and even for algebras which are not quantum groups. In particular, the dual canonical basis for the quantum group [O.sub.q][SL.sub.n](C) can be readily expressed in terms of Du's [7] dual canonical basis for the quantum polynomial ring A(n; q), which is not a quantum group.

In the above algebras, (dual) canonical bases are realted to several natural bases by transition matrices whose entries C[[q.sup.1/2], [q.sup.1/2]]-linear combinations of certain polynomials in Z[q]. These polynomials, which had appeared earlier in Kazhdan and Lusztig's work on Coxeter group and Hecke algebra representations [11] are called Kazhdan-Lusztig polynomials and R-polynomials. Even some such linear combinations, which can be defined in terms of single and double cosets of parabolic subgroups of Coxeter groups, had appeared earlier too. Specifically, those defined in terms of single parabolic cosets had appeared explicitly in the work of Douglass [5] and Deodhar [3]. These single parabolic analogs of Kazhdan-Lusztig and R- polynomials later assumed a prominent place in the literature, usually appearing in conjunction with single parabolic analogs of Hecke algebras. They appeared less frequently in conjunction with A(n; q), where they also arise naturally. On the other hand, the double parabolic analogs, some of which had appeared implicitly in the work of Curtis [2] even before the papers of Douglass and Deodhar, virtually escaped mention elsewhere in the literature.

We give a very brief overview of the natural appearances of double-parabolic Kazhdan-Lusztig and R-polynomials of type A in Hecke algebra modules and in the quantum polynomial ring A(n; q). In particular, we use the polynomials to give a new formulation of the dual canonical basis of A(n; q) and show that this is equivalent to Du's formulation [7].

In Sections 2-3, we consider the Hecke algebra Hn(q) and parabolic generalizations. In Section 4, we introduce the quantum polynomial ring A(n; q) and a duality which relates it to the earlier defined parabolic [H.sub.n](q)-modules. In Section 5, we apply the above duality to the zero-weight space (or immanant space) of A(n; q), to demonstrate some occurrences of (non-parabolic) Kazhdan-Lusztig and R-polynomials in this space. Some of these results appear (sadly) to be new. Finally, in Section 6, we discuss some brief appearances of double-parabolic Kazhdan-Lusztig and R-polynomials in the literature, and prove some additional results. In particular, we consider Du's formulation of the dual canonical basis of A(n; q) in terms of double-parabolic Kazhdan-Lusztig polynomials, and state an equivalent formulation which combines ordinary Kazhdan-Lusztig polynomials with row and column repetition within matrices. This result shows that one can understand all multigraded components of the dual canonical basis of A(n; q) in terms of zero-weight spaces of all quantum polynomial rings.

2 The symmetric group and Hecke algebra

The most basic results of Kazhdan and Lusztig [11] may be expressed in terms of the symmetric group [[??].sub.n] and Hecke algebra [H.sub.n](q) of type A.

Let [s.sub.1], ..., [s.sub.n-1] be the standard adjacent tranpositions generating [[??].sub.n], satisfying the relations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

A standard action of [[??].sub.n] on rearrangements of the word 1 ... n is defined by letting [s.sub.i] swap the letters in positions i and i + 1,

[s.sub.i] [omicron] [a.sub.i] ... [a.sub.n] = [a.sub.i] ... [a.sub.i+1][a.sub.i] ... [a.sub.n]. (2)

For each element [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we define the one-line notation of v to be the word [v.sub.1] ... [v.sub.n] = v [omicron] 1 ... n. Thus, denoting the identity permutation of [[??].sub.n] by e, the one-line notation of e is 12 ... n. Using this convention, the one-line notation of vw is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We will denote the one-line notation of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let ^(w) be the minimum length of an expression for w in terms of the generators, and let w0 denote the longest permutation in [[??].sub.n]. Let [less than or equal to] denote the Bruhat order on [[??].sub.n], i.e., v [less than or equal to] w if every reduced expression for w contains a reduced expression for v as a subword.

A deformation of [[??].sub.n] known as the Hecke algebra [H.sub.n](q) is the C[[q.sup.1/2], [q.sup.1/2]]-algebra generated by the set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of (modified) natural generators, subject to the relations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

(We follow [12].) If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a reduced expression for w [member of] [[??].sub.n] we define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [[??].sub.e] = 1. We shall call the elements {[[??].sub.w] | w [member of] [??]n} the natural basis of [H.sub.n](q) as a C[[q.sup.1/2], [q.sup.1/2]]-module. Specializing [H.sub.n] (q) at [q.sup.1/2] = 1, we recover the classical group algebra C[[[??].sub.n]] of the symmetric group. We remark that basis elements often denoted in the literature by {[T.sub.w] | w [member of] [[??].sub.n]} are related to our basis elements by [[??].sub.w] = [q.sup.-l(w)/2] [T.sub.w]. We will find it convenient to define the notation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

An involutive automorphism on [H.sub.n](q) commonly known as the bar involution is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

It is straightforward to check that the elements {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]} generate [H.sub.n](q). Expanding the basis {[bar.[[??].sub.w]] | w [member of] [S.sub.n]} in terms of the natural basis [11], we have

[bar.[??].sub.w] = [summation over v[less than or equal to]w] [q.sub.v,w][R.sub.v,w]([q.sup.-1])[[??].sub.v], (6)

where {[R.sub.v,w](q) | v, w [member of] [[??].sub.n]} are polynomials belonging to Z[q], and commonly called R- polynomials.

We call an element g of [H.sub.n](q) bar-invariant if it satisfies [bar.g] = g. Kazhdan and Lusztig showed [11] that [H.sub.n](q) has a unique basis of bar-invariant elements {[C'.sub.v](q) | v [member of] [[??].sub.n]} with

[C'.sub.v](q) [member of] [[??].sub.v] + [summation over (u<v)][q.sup.-1/2]Z[q.sup.-1/2][[??].sub.u]. (7)

Expanding [C'.sub.v](q) in terms of the natural basis, we have

[C'.sub.v](q) = [summation over (u[less than or equal to]v)][q.sup.-1.sub.u,v][P.sub.u,v](q)[[??].sub.v], (8)

where {[P.sub.u,v](q) | u, v [member of] [[??].sub.n]} are polynomials belonging to Z[q] (in fact, to N[q]). Call this basis the (sign-less) Kazhdan-Lusztig basis for [H.sub.n](q). Observe that (7) is equivalent to the conditions that [P.sub.u,u](q) = 1 for all u, [P.sub.u,v] (q) = 0 for u [??] v, and deg [P.sub.u,v](q) [less than or equal to] 1/2 (l(v) - l(u) - 1) for u < v.

Uniqueness of this basis [11, Sec. 2.2] follows from rewriting the condition [bar.[C'.sub.w](q)] = [C'.sub.w](q) as

[q.sub.u,w][P.sub.u,w] ([q.sup.-1]) - [q.sup.-1.sub.u,w][P.sub.u,w](q) = [summation over (u<v[less than or equal to]w] [q.sup.-1.sub.u,v][R.sub.u,v](q)[q.sup.-1.sub.u,w][P.sub.u,w](q) for all u [less than or equal to] w. (9)

In particular, there is a unique solution [P.sub.u,w] (q) satisfying this equation, when all other polynomials appearing are known. Existence of the basis [11, Sec. 2.2] follows from a recursive definition involving the function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

In particular, assuming that for some v we have defined bar-invariant elements {[C'.sub.u](q) | u [less than or equal to] v} satisfying the condition (7), we choose s so that v < sv and define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

One then verifies that [C'.sub.sv](q) also is bar-invariant and satisfies (7).

3 Parabolic generalizations of [H.sub.n](q)

Many of the concepts and results summarized in Section 2 generalize to a parabolic setting and were first stated by Curtis [2], Deodhar [3],[4], Douglass [5], and Du [8].

Let I and J be subsets of the standard generators of W = [[??].sub.r] and let [W.sub.I], [W.sub.J] be the corresponding parabolic subgroups of W. Define [W.sub.[phi]] = {e}. Let [W.sub.I]\W/[W.sub.J] be the set of double cosets of the form [W.sub.I] w [W.sub.J]. It is well known that each such double coset is an interval in the Bruhat order and thus has a unique minimal element and a unique maximal element. Let [W.sup.I,J.sub.+] be the set of maximal representatives of cosets in [W.sub.I]\W/[W.sub.J] and let [W.sup.I,J.sub.-] be the set of minimal coset representatives. Let [w.sup.I.sub.0] and [w.sup.J.sub.0] be the longest elements in [W.sub.I], [W.sub.J]. We have

[W.sup.I,J.sub.+] = {w | [s.sub.i]w < w, [ws.sub.j] < w for all [s.sub.i] [member of] I, [s.sub.j] [member of] J} (12)

We may define a Bruhat order on [W.sub.I]\W/[W.sub.J] in terms of maximal coset representatives v, w [member of] [W.sub.I,J.sub.+]: [W.sub.I]v[W.sub.J] [less than or equal to] [W.sub.I]w[W.sub.J] if and only if v [less than or equal to] w. Equivalently, by [5, Lem. 2.2], we may define this order in terms of minimal coset representatives.

For each permutation w [member of] [W.sup.I,J.sub.+], define the element

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Note that if I = [phi] and J = [phi], then each coset [W.sub.I]w[W.sub.J] in [W.sub.I]\W/[W.sub.J] is simply w [member of] [S.sub.n] and we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denote the submodule of [H.sub.n](q) spanned by these elements,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

The bar involution on [H.sub.n] (q) induces a bar involution on [H'.sub.I,J]. Curtis [2, Sec. 1] showed the elements {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]} to form a basis of [H'.sub.I,J], and Du [8, Sec. 1] showed that we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

where {[R.sup.I,J.sub.u,v](q)| u, v [member of] [W.sup.I,J.sub.+]} are polynomials belonging to Z[q]. This was probably the first mention in the literature of double-parabolic R-polynomials, although Deodhar [3, Sec. 2] and Douglass [5, Sec. 3] had earlier considered the single-parabolic cases in which I = [phi] or J = [phi]. It follows that double-parabolic R-polynomials are related to ordinary R-polynomials by

[R.sup.I,J.sub.u,w] (q) = [summation over (v[member of][W.sub.I]w[W.sub.J])] [R.sub.u,v] (q). (16)

Curtis [2, Thm. 1.10] showed that the set {[C'.sub.v](q)|v [member of] [W.sup.I,J.sub.+]} forms a bar-invariant basis of [H'.sub.I,J] and satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

In particular, he showed that the expansion of [C'.sub.v] (q) in terms of the natural basis of [H'.sub.I,J] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (18)

where {[P.sub.u,v] (q)|u, v [member of] [W.sup.I,J.sub.+]} are simply the Kazhdan-Lusztig polynomials indexed by pairs of permutations in [W.sup.I,J.sub.+]. To prove uniqueness, one rewrites the condition [bar.[C'.sub.w](q)] = [C'.sub.w] (q) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)

In particular, there is a unique solution [P.sub.u,w] (q) satisfying this equation, when all other polynomials appearing are known. In this context, the Kazhdan-Lusztig polynomials are sometimes called parabolic Kazhdan-Lusztig polynomials.

We remark that Deodhar [4] chose to name these polynomials (in the single parabolic cases) in terms of minimal coset representatives, rather than maximal coset representatives. As many other authors followed suit, one sees in the literature that the most common indexing of Kazhdan-Lusztig polynomials in a parabolic setting does not match the original indexing due to Kazhdan and Lusztig [11]. In particular, Deodhar [3], [4] associated the parameter -1 to the Kazhdan-Lusztig and R-polynomials above (when I = 0 and v, w [member of] [W.sup.0,J.sub.+]), denoting [P.sub.v,w] (q) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively. Douglas [5] associated the parameter u = q to the same polynomials, denoting them [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively.

4 The quantum polynomial ring

A second noncommutative ring called the quantum polynomial ring A(n; q), can be used to express the quantum group [O.sub.q] SL(n, C), the quantum coordinate ring of SL(n, C), as a quotient.

The ring A(n; q) is generated as an C[[q.sup.1/2], [q.sup.-1/2]]-algebra by [n.sup.2] variables x = ([x.sub.1,1], ..., [x.sub.n,n]) representing matrix entries, subject to the relations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (20)

for all indices i < j, k < l. We have the isomorphism [O.sub.q] SL(n, C) [congruent to] A(n; q)/([det.sub.q] (x) - 1), where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

is the quantum determinant. Specializing A(n; q) at [q.sup.1/2] = 1, we obtain the commutative polynomial ring C[[x.sub.1,1], ...,[ .sub.xn,n]].

As a C[[q.sup.1/2], [q.sup.1/2]]-module, A(n; q) is spanned by monomials in lexicographic order, and we can use the relations above to convert any other monomial to this standard form. A(n; q) has a natural grading by degree,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (22)

where [A.sub.r] (n; q) is the C[[q.sup.1/2], [q.sup.-1/2]]-span of all monomials of total degree r. Furthermore, the natural basis {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]} of A(n; q) is a disjoint union

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

of bases of the homogeneous components {[A.sub.r] (n; q)|r [greater than or equal to] 0}. We may further decompose each homogeneous component [A.sub.r] (n; q) by considering pairs (K, M) of multisets of r integers, written as weakly increasing sequences 1 [less than or equal to] [k.sub.1] [less than or equal to] ... [less than or equal to] [k.sub.r] [less than or equal to] n, and 1 [less than or equal to] [m.sub.1] [less than or equal to] ... [less than or equal to] [m.sub.r] [less than or equal to] n. This leads to the multigrading

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (24)

where the last direct sum is over pairs (K, M) of r-element multisets of [n], and [A.sub.K,M] (n; q) is the C[[q.sup.1/2], [q.sup.-1/2]]-span of monomials whose row indices and column indices (with multiplicity) are equal to the multisets K and M, respectively. Thus the graded component [A.sub.[n],[n]] (n; q) is the C[[q.sup.1/2], [q.sup.-1/2]]- submodule of A(n; q) spanned by the monomials

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (25)

Defining[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any u, v [member of] [[??].sub.n], we may express the above basis as {[x.sup.e,w]|w [member of] [[??].sub.n]}. We will call elements of this submodule (quantum) immanants. In particular, for any C[[q.sup.1/2], [q.sup.-1/2]]-linear function f : [H.sub.n](q) [right arrow] C[[q.sup.1/2], [q.sup.-1/2]], we define the quantum f -immanant

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)

More generally, [A.sub.K,M] (n; q) is the C[[q.sup.1/2], [q.sup.-1/2]]-submodule of A(n; q) spanned by the monomials

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (27)

where the generalized submatrix [x.sub.K,M] of x is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (28)

Note that variables in the monomials (27) do not necessarily appear in lexicographic order, e.g.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (29)

It is easy to see that the monomials {[([x.sub.K,M]).sup.u,v]|u, v [member of] [[??].sub.r]} satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (30)

An r-element multiset M = [m.sub.1] ... [m.sub.r] on [n], for any r, determines a subset I = [??](M) of generators

[??](M) = {[s.sub.i]|[m.sub.i] = [m.sub.i+1]} (31)

of [[??].sub.r]. Let K, M be r-element multisets of [n], and define subsets I = [??](K), J = [??](M) of generators of [[??].sub.r]. Since the dimension of [A.sub.K,M] (n; q) is less than n! (unless K = M = [n], or equivalently, if I = J = 0), the spanning set (27) is not in general linearly independent. We may construct a basis for [A.sub.K,M] (n; q) by using just one permutation w from each coset in [W.sub.I]\W/[W.sub.J]. In particular,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (32)

Thus for each pair (K, M) of r-element multisets of [n] satisfying [??](K) = I, [??](M) = J, we have dim [A.sub.K,M] (n; q) = dim [H'.sub.I,J]. Define the bilinear form <,> : [A.sub.K,M] (n; q) x [H'.sub.I,J] [right arrow] C[[q.sup.1/2], [q.sup.-1/2]] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (33)

where we assume that v, w belong to [W.sup.I,J.sub.+]. This pairing of elements in [A.sub.K,M] (n; q) with sums of elements in [H.sub.n](q) is the beginning of a striking similarity between the two algebras. We explore this similarity further in Sections 5-6.

5 Duality between [H.sub.n](q) and the immanant space [A.sub.[n],[n]] (n; q)

Many of the concepts and results summarized in Section 2 have analogs in the immanant space [A.sub.[n],[n]] (n; q) of A(n; q), but are not easily found in the literature.

Analogous to the bar involution on [H.sub.n](q) is an involution on A(n; q) again called the bar involution and again denoted g [??] [bar.g]. Following Brundan [1], we define this second bar involution by [bar.q] = [q.sup.-1], [[bar.x].sub.i,j] - [x.sub.i,j] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (34)

where [alpha](a) is the number of pairs i < j for which [a.sub.i] = [a.sub.j]. It is easy to see that the bar involution is well defined, i.e., that it respects the defining relations of A(n; q). We remark that the above bar involution differs by a power of q2 from those used by Du [8] and Zhang [16].

The restriction of this involution to [A.sub.[n],[n]] (n; q) may be described by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (35)

It is easy to see that the elements [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] form a basis of [A.sub.[n],[n]] (n; q). Expressing these elements in terms of the natural basis, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (36)

where {[S.sub.v,w] (q)|v, w [member of] [[??].sub.n]} belong to Z[q]. We will call these the S-polynomials.

Proposition 5.1 For all u, v [member of] [[??].sub.n] we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof: Omitted.

In fact, since [q.sup.-1.sub.u,v] [R.sub.u,v] (q) = [[epsilon].sub.u,v] [q.sup.-1.sub.u,v] [R.sub.u,v] (q), we have [S.sub.u,v) (q) = [R.sub.u,v] (q), but only the fact stated in the proposition will generalize nicely in Section 6. By the inversion formula [11, Sec. 3]

[summation over (u[less than or equal to]v[less than or equal to]w)] [[epsilon].sub.u,v] [S.sub.u,v] (q) [R.sub.v,w] (q) = [[delta].sub.u,w], (37)

the bar involutions are compatible with the bilinear form on [A.sub.[n],[n]] (n; q) x [H.sub.n] (q) defined in (33).

Corollary 5.2 For all u, v [member of] [[??].sub.n], we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Studying Lusztig's work [13] on canonical bases and other authors' work relating these to quantized Schur algebras and A(n; q), Du [7] defined a basis of [A.sub.[n],[n]] (n; q) consisting of bar-invariant elements which we shall denote {[Imm.sub.v] (x)|v [member of] [[??].sub.n]}, which satisfy

[Imm.sub.v] (x) [member of] [x.sup.e,v] + [summation over (w>v)] [q.sup.-1/2] Z[q.sup.-1/2] [x.sup.e,w]. (38)

Writing [Imm.sub.v] (x) in terms of the natural basis as

[Imm.sub.v] (x) [member of] [x.sup.e,v] + [summation over (w[greater than or equal to]v)] [[epsilon].sub.v,w] [q.sup.- 1.sub.v,w] [Q.sub.v,w](q)[x.sup.e,w]. (39)

we have that {[Q.sub.v,w] (q)|v, w [member of] W} are precisely the inverse Kazhdan-Lusztig polynomials

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (40)

defined in [11, Sec. 3], [12]. Note that [Imm.sub.v] (x) is the f -immanant (26) corresponding to the function [f.sub.v] : [[??].sub.w] [??] [[epsilon].sub.v,w] [q.sup.-1.sub.e,v] [Q.sub.v,w] (q), and that [Imm.sub.e](x) = [det.sub.q] (x). (See also definitions in [1], [16] and nonquantum analogs in [14].) Observe that Equation (38) is equivalent to the conditions that [Q.sub.u,u] (q) = 1 for all u, [Q.sub.u,v] (q) = 0 for u [not less than or equal to] v, and deg [Q.sub.u,v](q) [less than or equal to] 1/2 (l(v) - l(u) - 1) for u < v.

Uniqueness of this basis follows from rewriting the condition [bar.[Imm.sub.v](x)] = [Imm.sub.v] (x) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (41)

In particular, there is a unique solution [Q.sub.u,w] (q) satisfying this equation, when all other polynomials appearing are known. Existence of the basis follows from a recursive definition involving the function [mu] defined in (10), or simply from the definitions in Equations (39)-(40) and a computation verifying bar invariance.

By the inversion formula for Kazhdan-Lusztig polynomials [11, Sec. 3]

[summation over (u[less than or equal to]v[less than or equal to]w] [[epsilon].sub.u,v] [Q.sub.u,v] (q) [P.sub.v,w] (q) = [[delta].sub.u,w], (42)

we therefore have for all u, v [member of] [[??].sub.n] that

<[Imm.sub.u](x), [C'.sub.v] (q)> = [[delta].sub.u,v], (43)

and the basis of immanants {[Imm.sub.v](x)|v [member of] [[??].sub.n]} for [A.sub.[n],[n]] (n; q) is dual to the Kazhdan-Lusztig basis {[C'.sub.v] (q)|v [member of] [[??].sub.n]} of [H.sub.n](q).

6 Main results

Just as many properties of Hn(q) generalize to a parabolic setting, the analogs of these properties for the immanant space [A.sub.[n],[n]] (n; q) generalize to other multigraded components of A(n; q). We summarize these, paying particular attention to their role in defining the dual canonical basis of A(n; q), and providing a new, alternative formulation of this basis in Theorem 6.4.

Fix multisets K, M of [n] and use Equation (31) to define subsets I = [??](K), J = [??](M) of generators of [[??[log.sub.r]. It is easy to see that the elements {[([x.sub.K,M]).sup.e,v]|v [member of] [[??].sub.n]} form a basis of [A.sub.K,M] (n; q). Expressing these elements in terms of the natural basis, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (44)

where {[S.sup.I,J.sub.v,w] (q)|v, w [member of] [[??].sub.n]} belong to Z[q]. Call these the parabolic S-polynomials.

Double-parabolic S-polynomials are related to ordinary S-polynomials by the following identity.

Theorem 6.1 For all u, w [member of] [W.sup.I,J.sub.+] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (45)

Proof: Omitted.

These polynomials also may be expressed in terms of double-parabolic R-polynomials as follows.

Theorem 6.2 For all u, v [member of] [W.sup.I,J.sub.+] we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Furthermore, the matrix of double-parabolic S-polynomials inverts the matrix of double-parabolic R-polynomials in the sense that

[summation over (u[less than or equal to]v[less than or equal to]w)] [[epsilon].sub.u,v] [S.sup.I,J.sub.u,v] (q) [R.sup.I,J.sub.v,w] (q) = [[delta].sup.u,w]. (46)

Proof: Omitted.

Thus the two bar involutions are compatible with the bilinear form on [A.sub.K,M] (n; q) x [H'.sub.I,J] defined in (33).

Corollary 6.3 For all u, v [member of] [W.sup.I,J.sub.+], we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Du [7] formulates the dual canonical basis of A(n; q), as a union of bases of the multigraded components in Equation (24). The basis of the multigraded component [A.sub.K,M] (n; q) consists of bar-invariant elements {[Imm.sup.K,M.sub.v] (x)|v [member of] [W.sup.I,J.sub.+]} satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (47)

Writing [Imm.sup.K,M.sub.v] (x) in terms of the natural basis of [A.sub.K,M] (n; q) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (48)

Du showed that the polynomials {[Q.sub.v,w] (q)|v, w [member of] W} are alternating sums of inverse Kazhdan-Lusztig polynomials

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (49)

(See also [1], [16].) Observe that Equation (47) is equivalent to the conditions that [Q.sup.I,J.sub.u,u] (q) = 1 for all u [member of] [W.sup.I,J.sub.+], [Q.sup.I,J.sub.u,v] (q) = 0 for u [not less than or equal to] v, and deg [Q.sup.I,J.sub.u,v] and deg [Q.sup.I,J.sub.u,v] (q) [less than or equal to] 1/2(l(v) - l(u) - 1) for u < v.

Uniqueness of this basis follows from rewriting the condition [bar.[Imm.sup.K,M.sub.v](x)] = [Imm.sup.K,M.sub.v] (x) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (50)

In particular, there is a unique solution [Q.sup.I,J.sub.u,w] (q) satisfying this equation, when all other polynomials appearing are known. Existence of the basis follows from the definitions in Equations (48), (49) and a computation verifying bar-invariance. By the inversion formula [7, Sec. 1] for double-parabolic inverse Kazhdan-Lusztig polynomials

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (51)

we therefore have for all u, v [member of] [W.sup.I,J.sub.+] that

<[Imm.sup.K,M.sub.u] (x), [C'.sub.v] (q)> = [[delta].sub.u,v], (52)

and the subset of dual canonical basis elements {[Imm.sup.K,M.sub.v] (x)|v [member of] [W.sup.I,J.sub.+]} belonging to [A.sub.K,M] (n; q) is dual to the Kazhdan-Lusztig basis {[C'.sub.v] (q)|v [member of] [W.sup.I,J.sub.+]} of [H'.sub.I,J].

Using all of the above facts about double-parabolic Kazhdan-Lusztig, and S- polynomials, we can now state and prove our main result, which expresses dual canonical basis elements for any multigraded component of A(n; q) in terms of ordinary (inverse) Kazhdan-Lusztig polynomials.

Theorem 6.4 Fix multisets K, M and let I = [??](K). J = [??](M). For v [member of] [W.sup.I,J.sub.+] we have

[Imm.sup.K,M.sub.v] (x) = [Imm.sub.v] ([x.sub.K,M ]). (53)

Proof: Omitted.

This result generalizes the non-quantum immanant formulation in [15] of the dual canonical basis for C[[x.sub.1,1], ..., [x.sub.n,n]], and shows that one can express the dual canonical basis of A(n; q) in terms of zero- weight spaces of the quantum polynomial rings {A(r; q)|r [greater than or equal to] 0}.

We remark that Deodhar [3], [4] associated the parameter q to the inverse Kazhdan-Lusztig and S-polynomials above (when I = 0 and v, w [member of] [W.sup.0,J.sub.+]), denoting [Q.sub.v,w] (q) and [S.sup.0,J.sub.v,w] (q) by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively. Douglas [5] associated the parameter -1 to the same polynomials, denoting them [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively.

References

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Mark Skandera (1), Justin Lambright (2)

(1,2) Lehigh University Department of Mathematics, 14 East Packer Ave., Bethlehem, PA 18015, USA

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Author: | Skandera, Mark; Lambright, Justin |
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Publication: | DMTCS Proceedings |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Jan 1, 2009 |

Words: | 5382 |

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