SUPERNOMIAL COEFFICIENTS, BAILEY'S LEMMA AND ROGERS ...

16 S. O. WARNAARFinally we note that a level-N A n−1 Euler identity is obtained if we sum(6.5) by application of (6.4),∑ ∑ɛ(σ)q 1 ∑ n2 i=1 (nk i−2σ i )k i + 1 N (nk i−σ i +i)(6.8)2 C nk−σ+ρ (q) = 1.σ∈S n|k|=07. Supernomial identities and beyond?So far we have given two applications of theorem 3.2, based on the initial conditionidentities (4.1) and (4.4). Many more identities can however be derived.In ref. [37] an infinite hierarchy of A 2 supernomial identities was conjectured,of which (4.1) is the first instance. Taking this conjectured hierarchy as inputto theorem 3.2 leads to a doubly-infinite family of A 2 q-series identities. Wewill not carry out this programme in full here, but shall instead make someintriguing observations concerning some of the identities that may be derived.In fact, since all will be conjectural, we shall present our speculations in a moregeneral A n−1 setting.First we need the conjecture of [37] (equation (9.2) with q → 1/q and N = 1).Using the tensor-product notation of the previous section we define for allintegers p ≥ n,F p,L (q) = q LCL2(p−n) ∑(q) CLµq 1 2 η(C⊗C−1 )η[ µ + ηHere L ∈ Z n−1+ , C ⊗ C −1 is the tensor product of the A n−1 and inverse A p−n−1Cartan matrices, and the sum is over µ ∈ Z n−1 ⊗ Z p−n−1 , with entries µ (a)j ,a = 1, . . . , n − 1, j = 1, . . . , p − n − 1, such that(7.1)(C −1 ⊗ I)µ ∈ Z p−n−1 .The vector η is determined by L and µ through the relation(7.2)As special cases we haveandη = L ⊗ e 1 − (C −1 ⊗ C)µ.F n,L (q) = δ L1 ,0 . . . δ Ln−1 ,0F n+1,L (q) = q 1 2 LCL(q) CL.Conjecture 7.1. For n ≥ 2, p ≥ n, L ∈ Z n−1+ and k ∈ Z n such that |k| = 0,∑ ∑ɛ(σ)q 1 ∑ n2 i=1 (pk i−2σ i )k(7.3)iS(L, pk − σ + ρ) = F p,L (q).σ∈S n|k|=0µ].