M\"obius Polynomials and Splitting Algebras of Direct Products of Posets

TL;DR

This paper derives a formula for the M"obius polynomial of the direct product of ranked posets, enabling calculations of Hilbert series and trace functions for splitting algebras of specific posets like Boolean algebras.

Contribution

It introduces a new formula linking the M"obius polynomial of product posets to their factors, advancing the understanding of splitting algebra invariants.

Findings

Derived a formula for M"obius polynomial of product posets

Calculated Hilbert series for splitting algebras of Boolean algebra

Computed graded trace generating functions for posets of factors of n

Abstract

In this paper, we will study the M\"obius polynomial, an invariant of ranked posets that arises in the study of splitting algebras. We will present a formula for the M\"obius polynomial of the direct product of posets in terms of the M\"obius polynomials of the factors. We will then use this formula to calculate Hilbert series and graded trace generating functions associated to the splitting algebras of the Boolean algebra and the poset of factors of a natural number n.