A Note on Quantum Immanants and the Cycle Basis of the Quantum Permutation Space

There are many combinatorial expressions for evaluating characters of the Hecke algebra of type A. However, with rare exceptions, they give simple results only for permutations that have minimal length in their conjugacy class. For other permutations, a recursive formula has to be applied. Consequently, quantum immanants are complicated objects when expressed in the standard basis of the quantum permutation space. In this paper, we introduce another natural basis of the quantum permutation space, and we prove that coefficients of quantum immanants in this basis are class functions. 1. The symmetric group and immanants Denote by Sn the symmetric group of n, i.e. the group of permutations of the set {1, . . . , n}. We write permutations in the one-line notation: v = v1v2 · · · vn means that v(i) = vi. We multiply permutations from the right: 24315 · 53241 = 53412. We will often use the cycle notation 24315 = (124)(35). We will always write the smallest element of the cycle first, and order the cycles so that the first elements form an increasing sequence. We define the cycle type μ(v) as the sequence of lengths of these cycles. Note that it is a composition, not a partition; permutations (124)(35) and (14)(253) have a different cycle type. An inversion of a permutation v is a pair (i, j) satisfying i < j and vi > vj. Denote by inv(v) the number of inversions of v. We denote the identity permutation by id. The symmetric group Sn is generated by simple transpositions si = (i, i+1), 1 ≤ i ≤ n−1, which satisfy the relations si = 1 for i = 1, . . . , n− 1, sisi+1si = si+1sisi+1 if |i− j| = 1, sisj = sjsi if |i− j| ≥ 2. An expression v = si1si2 · · · sik , 1 ≤ ij ≤ n−1, is reduced if it is the shortest such expression for v, and we have k = inv(v). We call k the length of v. All reduced expressions contain the same generators, see [BB05, Corollary 1.4.8 (ii)]. A (virtual) character of a group G is a linear function χ : G → C for which χ(ab) = χ(ba) for all a, b ∈ G. For example, the trace of a representation ρ : G → GLn is a character. The simplest character is the trivial character η(v) = 1. In the symmetric group, another important character is the sign character 2(v) = (−1)inv(v). Choose commutative variables xij, 1 ≤ i, j ≤ n. Denote by An the vector space of all polynomials in xij generated by monomials of the form xv = x1v1x2v2 · · · xnvn for a permutation v ∈ Sn, and call An the permutation space. We will also use notation 1