Presentations of Principal Subspaces of Higher Level Standard A2(2)${A}_{2}^{(2)}$-Modules

Abstract

We study the principal subspaces of higher level standard A(2) 2modules, extending earlier work in the level one case, by Calinescu, Lepowsky and Milas. We prove natural presentations of principal subspaces and also of certain related spaces. By using these presentations we obtain exact sequences, which yield recursions satisfied by the characters of the principal subspaces and related spaces. We conjecture a formula for a specialized character of the principal subspace, given by the Nahm sum of the inverse of the tadpole Cartan matrix.

Principal subspaces of basic modules for twisted affine Lie algebras, q-series multisums, and Nandi’s identities

“…, a r ; q) = θ(a 1 ; q)θ(a 2 ; q) • • • (a r ; q). (14) Algebraic Combinatorics, Vol. 6 #6 (2023) Principal subspaces and q-series multisums 2.3.…”

“…Vertex-algebraic background. We recall certain details relevant to this work from the vertex-algebraic constructions found in the works of Calinescu, Lepowsky, Milas, Penn, Webb, and the fourth author [12,14,35,36,37]. Suppose (15)…”

“…In particular, the graded dimensions of principal subspaces are interesting due to their connection to various partition identities and recursions they satisfy. The study of principal subspaces for standard modules of twisted affine Lie algebras was initiated by Calinescu, Lepowsky, and Milas [12], and further developed in works by Calinescu, Milas, Penn, and the fourth author [13,35,36,14]. The multigraded dimensions for principal subspaces of basic (i.e., the standard module with highest weight Λ 0 , see, for example, Carter's book [18, P. 508]) modules for twisted affine Lie algebras are wellknown, and have been studied in several of those papers [13,35,36].…”

“…This work grew out of the following observations: Calinescu, Milas, and Penn [13] studied the graded dimension of the principal subspace of the basic A (2) 2n module and showed that it is given by (2) m∈(Z ⩾0 ) n q m t Tn m 2 (q; q) m1 • • • (q; q) mn where T n is the Cartan matrix of the tadpole Dynkin diagram. Meanwhile, Calinescu, Penn, and the fourth author [14] conjectured that the graded dimension of the principal subspace of the standard A…”

Ghost series and a motivated proof of the Andrews–Bressoud identities

“…Lepowsky and A. Milas, and is currently under investigation. Similar categorifications of the Rogers-Ramanujan, Rogers-Selberg and Euler recursions, as well as of new recursions, using untwisted and unrelativized intertwining operators, have already been successfully carried out in [CLM1], [CLM2], [Cal1], [Cal2], [CalLM1]- [CalLM4] and [Sa].…”