Relations between the Ehrhart polynomial, the heat kernel and Sylvester waves

Abstract

I show for the specific case of the scalar field spectrum on regular tessellations of the sphere that the first two terms of the heat--kernel expansion are related to the first two terms of the Ehrhart (quasi)polynomial. In trying to make this relation precise, I consider degeneracies as partition denumerants and show the connection of the group theory expressions with Popoviciu's theorem and with the notion of Sylvester waves. General denumerants are considered and the first wave, i.e. the polynomial part, is written using the A-genus multiplicative sequence. It is pointed out that Sylvester in effect did the same thing and that he had also obtained Ehrhart reciprocity. I derive an algebraically neat form for the second wave which involves the combination of two multiplicative sequences.