Untitled

Abstract

In the 1990s, Cherednik introduced the double affine Hecke algebra [2]. In terms of the polynomial representation of the algebra, Macdonald's conjecture was solved in [3,4] for reduced root systems (and in [12, 13] for nonreduced (C ∨ , C) case). For the root system of type A, a classification of irreducible representations of a certain class is given in [6,14,15]. In [1], it is shown that finite-dimensional quotients of the polynomial representation by the kernel of some degenerate bilinear form are irreducible.The double affine Hecke algebra H n of type GL n is an associative algebra with two parameters t, q generated by X i , Y i , and T j (1 ≤ i ≤ n, 1 ≤ j ≤ n − 1) with some relations. The algebra H n has a basic representation U on the ring of n-variable Laurent polynomials. The representation U is irreducible and Y-semisimple, namely, the operators Y i are simultaneously diagonalizable on U. The nonsymmetric Macdonald polynomial is defined to be a monic simultaneous eigenvector for Y i .In this paper, we specialize the parameters at t k+1 q r−1 = 1 for 1 ≤ k ≤ n − 1 and 2 ≤ r. To be precise, introduce a new parameter u and specialize (t, q) = u (r−1)/M , τu −(k+1)/M .(1.1)Here, M is the greatest common divisor of (k + 1, r − 1) and τ = exp(2π √ −1/(r − 1)). We http://imrn.oxfordjournals.org/ Downloaded from the space of Laurent polynomials satisfying the wheel condition (1.3).We now state the main theorem in this paper. Define B (k,r) = {λ ∈ Z n ; for any 1 ≤ a ≤ n − k, λ ia − λ i a+k ≥ r, or λ ia − λ i a+k = r − 1, i a < i a+k }. Here, we determine the index (i 1 , . . . , i n ) = w · (1, . . . , n) using the shortest element w ∈ W = S n such that λ = w · λ + , where λ + is the dominant element in the orbit Wλ. The result is as follows. at Biology Library on March 10, 2015 http://imrn.oxfordjournals.org/ Downloaded from Subrepresentations in the Polynomial Representation 1719We write f(t ±ρ(λ) q ±λ ) = f(t ±ρ(λ) 1 q ±λ 1 , . . . , t ±ρ(λ)n q ±λn ). The nonsymmetric Macdonald polynomial E λ is defined to be the monic Y-eigenvector: f(Y)E λ = f t ρ(λ) q λ E λ for any f ∈ U,(2.6)The nonsymmetric Macdonald polynomials form a basis of U.

Theta function solutions of the quantum Knizhnik–Zamolodchikov–Bernard equation for a face model

“…3.4) which satisfy the wheel condition (3.1). The corresponding problem in the trigonometric case was solved by Kasatani in [27], leading to a a connection to nonsymmetric Macdonald polynomials [28].…”

Spin-singlet quantum Hall states and Jack polynomials with a prescribed symmetry

“…Jack polynomials, both symmetric [22,23] and non-symmetric [17], enjoy (k, r) clustering properties at the following negative value of the coupling…”

“…although in the following we focus on n = 2. For (k, r)-admissible partitions S N ↑ ⊗ S N ↓ symmetric Jack polynomials are well defined at α = − k+1 r−1 , and enjoy the clustering properties inherited from the wheel condition of [17,18]. In particular…”

“…Moreover this set of wavefunctions (in the sense of ( 5)) is stable under the action of the total SU(2) spin generators S ± = i S ± i and S z = i S z i . This property is inherited from the stability of the ideal of non-symmetric Jack polynomials at α = −(k + 1)/(r − 1) with (k, r) admissible partitions under the full group of permutations S N [17].…”

Grassmann–Grassmann conormal varieties, integrability, and plane partitions

“…, s σ(a) )(we postpone to §5.2 the definition of the weights of the B, C variables of the ring, which we do not need for now). Finally, for x any linear combination of such monomials, we define in(x) to be the sum of monomials of x for which the function wt is minimal.If we let P denote the space of Plücker relations, cf(12), viewed as linear forms on such monomials, namely P = span p s 1 . .…”