We introduce hierarchies of difference equations (referred to as nT -systems) associated to the action of a (centrally extended, completed) infinite matrix group GL (n) ∞ on n-component fermionic Fock space. The solutions are given by matrix elements (τ -functions) for this action. We show that the τfunctions of type nT satisfy bilinear equations of length 3, 4, . . . , n + 1. The 2T -system is, after a change of variables, the usual 3 term T -system of type A.Restriction from GL (n)∞ to a subgroup isomorphic to the loop group LGLn, defines nQ-systems, studied earlier in [1] by the present authors for n = 2, 3.
Q-systems and T -systems are systems of integrable difference equations that have recently attracted much attention, and have wide applications in representation theory and statistical mechanics. We show that certain τ -functions, given as matrix elements of the action of the loop group of GL 2 on two-component fermionic Fock space, give solutions of a Q-system. An obvious generalization using the loop group of GL 3 acting on threecomponent fermionic Fock space leads to a new system of 4 difference equations.
We briefly describe what tau-functions in integrable systems are. We then define a collection of tau-functions given as matrix elements for the action of GL 2 on two-component Fermionic Fock space. These tau-functions are solutions to a discrete integrable system called a Q-system.We can prove that our tau-functions satisfy Q-system relations by applying the famous "Desnanot-Jacobi identity" or by using "connection matrices", the latter of which gives rise to orthogonal polynomials. In this paper, we will provide the background information required for computing these tau-functions and obtaining the connection matrices and will then use the connection matrices to derive our difference relations and to find orthogonal polynomials.We generalize the above by considering tau-functions that are matrix elements for the action of GL 3 on three-component Fermionic Fock space, and discuss the new system of discrete equations that they satisfy. We will show how to use the connection matrices in this case to obtain "multiple orthogonal polynomials of type II".2010 Mathematics Subject Classification. 17B80. Thanks to Rinat Kedem and Philippe Di Francesco for their helpful comments.
We give some general results about the generators and relations for the higher level Zhu algebras for a vertex operator algebra. In particular, for any element u in a vertex operator algebra V , such that u has weight greater than or equal to −n for n ∈ N, we prove a recursion relation in the nth level Zhu algebra A n (V ) and give a closed formula for this relation. We use this and other properties of A n (V ) to reduce the modes of u that appear in the generators for A n (V ) as long as u ∈ V has certain properties (properties that apply, for instance, to the conformal vector for any vertex operator algebra or if u generates a Heisenberg vertex subalgebra), and we then prove further relations in A n (V ) involving such an element u. We present general techniques that can be applied once a set of reasonable generators is determined for A n (V ) to aid in determining the relations of those generators, such as using the relations of those generators in the lower level Zhu algebras and the zero mode actions on V -modules induced from those lower level Zhu algebras. We prove that the condition that (L(−1) + L(0))v acts as zero in A n (V ) for n ∈ Z + and for all v in V is a necessary added condition in the definition of the Zhu algebra at level higher than zero. We discuss how these results on generators and relations apply to the level n Zhu algebras for the Heisenberg vertex operator algebra and the Virasoro vertex operator algebras at any level n ∈ N.
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