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  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.