A criterion for polynomial growth of a-free two-peak posets of tame prinjective type*

Abstract

It is well known from the results of L, A. Nazarova and A. G. Zavadskij [18], [19], see also [25, Chapter 15], that a poset J with one maximal element is of tame prinjective type and of polynomial growth if and only if J does not contain neither any of the Nazarova's hypercritical posets (1,1,1,1,1)*, (1,1,1,2)*,(2,2,3)*, (1,3,4)*,(W,5)*,(1,2,6)* nor the Nazarova-Zavadskij poset (NZ)* (see Table 1 below). In the present paper we extend this result to a class of posets with two maximal elements. We show that Ã-free poset with two maximal elements is of tame representation type and of polynomial growth if and only if the Tits quadratic form qs → Z (see (1.7) below) associated with J is weakly non-negative and J does not contain any of the six posets listed in Table 1 as a peak subposet.