computation of the Hall coefficient g(q)[('6,4,2)(,42)(,4,2)]

Let L be a uniserial ring of length n, with maximal ideal r , and finite residue field Λ/ r . We consider Λ-modules which possess a finite composition series. We note that a Λ-module has the form B ≅ ⨁i=1m Λ/ rli , where the type of B is the partition l = ( l1,&ldots;,lm ) denoted by t(B). For Λ-modules A, B, C with t(A) = m , t(B) = l , t(C) = n , if A ⊆ B, and B/A ≅ C, we define GBAC = |{U ⊆ B : U ≅ A and B/U ≅ C}|. We show that GBAC = MonoA,B,C Aut A = | S (A, B, C)/∼| = glmn (q), where |Λ/ r | = q, and the last equality comes from evaluating the Hall polynomial glmn (t) ∈ Z [t] at q, as stated in Hall's Theorem. We note that GBAC make up the coefficients of the Hall algebra. We provide a proof that the Hall algebra is a commutative and associative ring. Using the property of associativity of the Hall algebra and I. G. MacDonald's formula: glb1l =qnl -nb-n 1li≥ 1l'i -b'i,b' i-l'i+1 q-1 we develop a procedure to generate arbitrary Hall polynomials and we compute g6,4,2 4,24,2 (q).