By Kenneth B Stolarsky

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Qn-i-1 _l)qi(n-m) i= 1 x (qm-i·• 1 _ 1) (q m-i+ 2-l) ... (qm-1) 1 2 + (-l)m(q-1) (q -1) · · · (qn-m- -1) . 2 (q-l)(q -1) ... (q 1 -1) We compute the difference E.. : 1. Observe that qi(n-m-1) (qm-i-t-2_ 1 ) (qm-i+3 _l) ... (qm+l _1) _ q i(n-m) (qm-i-1 l _l)(qm-i+2 _l) ... (qm-1) q i(n-m-1) (qm-i+2 _l) (qm-i+ 3-1) ... (qm-1) (qi-1) We have E"(n,m+l) - E"(n,m) = m+l l i= 1 (-l)i+l (q-l)(q 2 -1) ··· (qn-i- 1 -1) PARTIALLY ORDERED SETS AND HOMOLOGY = qn-m-1 ((q-l)(q2-l) ... (qn-2_1) - E"(n-1, m)) This implies clearly the desired formula by induction on n-1 m.

A (V) satisfying (8) must be zero. 3 In this section we shall assume that dim V 2. Let Pol q- 2 (V) be the set of homogeneous polynomial functions V .... F of degree (q-2). It = is clear that Polq_ 2 (V) is an F -vector space of dimension (q-1). We shall describe now two F-vector spaces which are canonically isomorphic to Polq_ 2 (V) but whose definitions are somewhat more geometric. pace of all functions f which associate to any one-dimensional linear subspace V1 of V a vector f(V1 ) € V1 such that l f(V1 ) = 0 (sum in V).

Orbits l -> I permutations of 1, 2, .. ·, nl is bijective. We need an extension of this result to the affine case. Such an extension has been found by Solomon [12] but we shall need a somewhat different approach. The affine space E can be regarded as an affine hyperplane not containing the origin in an (ftl)-dimensional vector space V. Let H be the unique linear hyperplane in V which is parallel to E. Note that 32 THE DISCRETE SERIES OF GLn OVER A FINITE FIELD Aff(E) can be regarded as the set of all t c GL(V) such that t(E) = E.