Graded trace generating functions of Aut(ΓKn) acting on A(ΓKn)

Abstract

In this paper, we describe the algebras [Formula: see text] associated with the Hasse graph of the poset of the faces of the complete graph at n vertices [Formula: see text]. Present the relations that define this algebra, calculate the Hilbert series of quadratic grade algebra [Formula: see text] and the graded trace generating functions of [Formula: see text] acting on [Formula: see text] and show that [Formula: see text] is a Koszul algebra. The methodology adopted in this paper consists of building the algebra [Formula: see text] using [I. Gelfand, V. Retakh, S. Serconek and R. L. Wilson, On a class of algebras associated to directed graphs, Selecta Math. 11(2) (2005) 281], giving a presentation based on what was established by [V. Retakh, S. Serconek and R. L. Wilson, On a class of koszul algebras associated to directed graphs, J. Algebra 304(2) (2006) 1114–1129] in terms of the vertices of the graph [Formula: see text]. After that, we use [V. Retakh, S. Serconek and R. L. Wilson, Hilbert series of algebras associated to directed graphs, J. Algebra 312(1) (2007) 142–151] to determine the Hilbert series of [Formula: see text] and use [9] again to show that this algebra is a Koszul algebra. We use [J. Caldeira, A. De Souza Lima and J. Eder Salvador De Vasconcelos, Representations of automorphism groups of algebras associated to star polygons, J. Algebra Appl. 18(10) (2019) 1950197; C. Duffy, Representations of [Formula: see text] acting on homogeneous components of [Formula: see text] and [Formula: see text], Adv. Appl. Math. 42(1) (2009) 94–122] to determine the generating functions of the graduated trace of [Formula: see text] acting on [Formula: see text].