Elliptic Littlewood identities (Q439064)

Among the identities satisfied by the interpolation functions is an analogue of the Cauchy identity which for Macdonald polynomials states that NEWLINE\[NEWLINE \sum_\mu P_\mu(x_1,\dotsc,x_n;q,t)P_{\mu'}(y_1,\dotsc,y_m;t,q)=\prod_{1\leq i\leq n,1\leq j\leq m}(1+x_iy_j). NEWLINE\]NEWLINE Macdonald also proved an analogue of the Littlewood identity for Macdonald polynomials NEWLINE\[NEWLINE \sum_\mu c_\mu(q,t)P_{\mu^2}(x_1,\dotsc,x_n;q,t)=\prod_{1\leq i<j\leq n}\frac{(tx_ix_j;q)}{(x_ix_j;q)}, NEWLINE\]NEWLINE where \(P_\mu\) is a Macdonald polynomial, \(\mu^2\) denotes the partition with parts \((\mu^2)_i=\mu_{\lceil i/2\rceil}\), \((x;q)= \prod_{k\geq0}(1-q^kx)\), and the coefficients \(c_\mu(q,t)\) are given by an explicit product.NEWLINENEWLINEIn this paper, analogues for elliptic interpolation functions of Macdonald's version of the Littlewood identity for skew Macdonald polynomials, in the process developing an interpolation of general elliptic ``hypergeometric'' sums as skew interpolation functions are proved. One such analogues has an interpolation as a ``vanishing integral'', generalizing a result of \textit{E. M. Rains} and \textit{M. Vazirani} [Transform. Groups 12, No. 4, 725--759 (2007; Zbl 1202.33027)]. The structure of this analogue gives sufficient insight in order to enable us to conjecture elliptic versions of most of the other vanishing integrals of Rains and Vazirani, as well. We are thus led to ten conjectures (L1--L3 and Q1--Q7), each of which can be viewed as a multivariate quadratic transformation and is proved in special cases.