By John G. Hocking

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Analogously, we define columnordered and column-strict arrays. Further, we define a lexicographic order on N+ × N+ via (i, j) < (i , j ) ⇐⇒ either j < j , or j = j and i > i . Given a row-strict array A of shape α, we define d(A) = |{(x, y) ∈ α × α| y < x and A(x) < A(y) < A(x→ )}|. 45. We have q d(A) bλµ = A where A ranges over all row-strict arrays of shape µ and weight λ . We will provide a proof in the next lecture. In this lecture, we will analyze consequences of the theorem. 46. Given a row-strict array of shape α and weight β, we introduce a {0, 1}matrix which has entry 1 at the positions {(i, A(x))} where x = (i, j) runs over all x ∈ α, and entry 0 elsewhere.

To show (1) we work with n variables x1 , . . , xn . Denote by er symmetric polynomials in x1 , . . , xk−1 , xk+1 , . . , xn . They satisfy the elementary n−1 E (k) er(k) tr = (t) = r=0 (1 + xi t). i=k On the other hand, we have n (1 − xi t)−1 = H(t) = i=1 hr (x1 , . . , xn )tr . r≥0 For α ∈ Nn , we compare coefficients of tαi in the equality H(t)E (k) (−t) = (1 − xk t)−1 to obtain the equations n (k) hαi −n+j (−1)n−j en−j = xαi . j=1 We may express this set of equations as a matrix identity Hα M = Aα (k) where Hα = (hαi −n+j )i,j , M = ((−1)n−j en−j )j,k , and Aα = (xαk i )i,k .

Since the G is π0 -finite, the resulting decomposition of X has only finitely many components. It therefore suffices to show that each of these components has finitely many isomorphism classes so that we have reduced to the case 57 when B and C (and A are connected). In other words, given group homomorphisms ϕ : G → H, and ψ : G → H, we have to show that (2) X = BG ×BH BG has finitely many isomorphism classes, assuming that the map Bϕ : BG → BH is finite. As we have seen, the 2-fiber of Bϕ : BG → BH is the action groupoid corresponding to the G-action on H via ϕ.