We use a weight-preserving, sign-reversing involution to find a combinatorial expansion of ∆e k en at q = 1 in terms of the elementary symmetric function basis. We then use a weightpreserving bijection to prove the Delta Conjecture at q = 1. The method of proof provides a variety of structures which can compute the inner product of ∆e k en|q=1 with any symmetric function.
Let (x1, . . . , xn, y1, . . . , yn) be a list of 2n commuting variables, (θ1, . . . , θn, ξ1, . . . , ξn) be a list of 2n anticommuting variables, and C[xn, yn] ⊗ ∧{θn, ξ n } be the algebra generated by these variables. D'Adderio, Iraci, and Vanden Wyngaerd introduced the Theta operators on the ring of symmetric functions and used them to conjecture a formula for the quadruply-graded Snisomorphism type of C[xn, yn] ⊗ ∧{θn, ξ n }/I where I is the ideal generated by Sn-invariants with vanishing constant term. We prove their conjecture in the 'purely fermionic setting' obtained by setting the commuting variables equal xi, yi equal to zero.
In [10], the authors introduced tiered trees to define combinatorial objects counting absolutely indecomposable representations of certain quivers and torus orbits on certain homogeneous varieties. In this paper, we use Theta operators, introduced in [6], to give a symmetric function formula that enumerates these trees. We then formulate a general conjecture that extends this result, a special case of which might give some insight about how to formulate a unified Delta conjecture [20].
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