Hankel Pfaffians, Discriminants and Kazhdan-Lusztig bases

We use Kazhdan-Lusztig bases of representations of the symmetric group to express Pfaffians with entries (ai−aj)hi+j . In the case where the parameters ai are specialized to successive powers of q, and the hi are complete functions, we obtain the q-discriminant. 1 ’ ooo oo o ’ ooo oo o ’ ooo oo o ’ ooo oo o ’ ooo oo o ’ ooo oo o ’ ooo oo o ’ ooo oo o ’ ooo oo o ’ ooo oo o ’ ooo oo o ’ ooo oo o ’ ooo oo o ’ ooo oo o ’ ooo oo o ’ ooo oo o Hankel matrices are matrices constant along anti-diagonals. A prototype is M = ∣∣∣hi+j∣∣∣ i,j=1...n , with indeterminates hi in a commutative ring. With one more set of indeterminates ai, and an integer k ∈ Z, one defines the Hankel Pfaffian Pf(a,h, n, k) to be the Pfaffian of the antisymmetric matrix M(a,h, n, k) of order 2n with entries (ai − aj)hi+j−3+k . This is the Pfaffian that we shall study in this text. Such Pfaffians with ai = i or ai = q i and special hi have been considered by Ishikawa, Tagawa, Zeng [5]. Hankel matrices, when the hi are identified with complete functions of an alphabet of cardinality n, are related to resultants, Bezoutians, orthogonal polynomials, continued fractions, etc [10]. We show similarly in section 2 and section 5 that Hankel Pfaffians in complete functions allow to express resultants, Bezoutians, q-discriminants, and give several determinantal expressions of such Pfaffians. The Hankel Pfaffian Pf(a,h, n, k) can be studied by mere algebraic manipulations, this is what we do in section 2. However, it is much more fruitful to use the action of the symmetric group on the indeterminates ai. In [11], we have shown how to diagonalize Pfaffians using Young’s idempotents. In the present case, it is more convenient to use the bases of Kazhdan and Lusztig[7]. Theorem 13 shows, indeed, that Pf(a,h, n, k) is diagonal in a pair of adjoint Kazhdan-Lusztig bases. article based on a talk given at the First Euro-Korean Conference on Groups and Related topics, March 2011, Postech University.