POLYNOMIALS RELATED TO HALL NUMBERS

Abstract

Let Λ be a finite dimensional algebra of finite representation type over a finite field k. For any modules A, B and Pin mod Λ with P projective, we prove that there exists a polynomial ϕ ^B _(P)Λ over Z whose evaluation at |E| for any conservative finite field extension E of Λ is the sum of Hall numbers F ^{B E} _{C ^E A ^E} where C ^E runs through isoclasses in mod Λ ^E and P ^E is the projective cover of C ^E . As a consequence of this result and its dual version, Hall polynomials ϕ ^E _CA exist when C or A is semisimple. As applications of the main result, we obtain the existence of Hall polynomials for Nakayama algebras and some selfinjective algebras.

^*Supported by the Volkswagen Foundation.

ACKNOWLEDGMENTS

This work was done during the stay at Universität Bielefeld. The auther would like to thank Professor Claus Michael Ringel for his guidance; thank Professor Liangang Peng for his encouragement. Thanks also go to the referee for his or her helpful comments on the manuscript.

Notes

^*Supported by the Volkswagen Foundation.