From Partition Identities to a Combinatorial Approach to Explicit Satake Inversion

Abstract

In this paper, we provide combinatorial proofs for certain partition identities which arise naturally in the context of Langlands' beyond endoscopy proposal. These partition identities motivate an explicit plethysm expansion of Sym j pSym k V q for GL 2 in the case k " 3. We compute the plethysm explicitly for the cases k " 3, 4. Moreover, we use these expansions to explicitly compute the basic function attached to the symmetric power L-function of GL 2 for these two cases.

A Family of Theta-Function Identities Based upon Combinatorial Partition Identities Related to Jacobi’s Triple-Product Identity

“…In particular, Chaudhary et al generalized several known results on character formulas (see [22]), Roger-Ramanujan type identities (see [19]), Eisenstein series, the Ramanujan-Göllnitz-Gordon continued fraction (see [20]), the 3-dissection property (see [18]), Ramanujan's modular equations of degrees 3, 7, and 9 (see [16,17]), and so on, by using combinatorial partition-theoretic identities. An interesting recent investigation on the subject of combinatorial partition-theoretic identities by Hahn et al [25] is also worth mentioning in this connection.…”

“…Various extensions and generalizations of partition-theoretic identities and other q-identities, which we have investigated in this paper, as well as their connections with combinatorial partition-theoretic identities, can be found in several recent works (see, for example, [31,34,35]). The demonstrations in some of these recent developments are also based upon their combinatorial interpretations and generating functions (see also [25]).…”

A family of theta-function identities based upon Rα,Rβ and Rm-functions related to Jacobi’s triple-product identity

“…The list of citations, which we have included in this article, is believed to be potentially useful for indicating some of the directions for further researches and related developments on the subject-matter which we have dealt with here. In particular, the recent works by Adiga et al (see [1]- [3]), Cao et al [9], Chaudhary et al (see [10] to [16]), Hahn et al [17], and Srivastava et al (see [26], [31]- [33]) are worth mentioning here.…”

“…In particular, Chaudhary et al generalized several known results on character formulas (see [13]), Ramanujan's modular equations of degrees 3, 7 and 9 (see [10] and [11]), and so on, by using combinatorial partition-theoretic identities. An interesting recent investigation on the subject of combinatorial partition-theoretic identities by Hahn et al [17] is also worth mentioning in this connection.…”

On relationships between q-products identities, Ralpha, Rbeta and Rm functions related to Jacobi's triple-product identity

“…Thereafter, several new advancements and generalizations of the existing results were made in regard to combinatorial partition-theoretic identities (see, for example, [15] to [17]). An interesting recent investigation on the subject of combinatorial partition-theoretic identities by Hahn et al [9] is also worth mentioning in this connection. In this paper, our main objective is to establish a set of two new identities which depict the inter-relationships in terms of R α , R β and R m functions along with q-product identities.…”