By Tobias Dyckerhoff

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**Additional resources for Hall Algebras [Lecture notes]**

**Example text**

From the calculation of the local 2-fibers of F in the proof of the above Theorem, it follows that the value of the function F! ◦ G∗ (✶([A],[A ]) ) at B ∈ X1 is given by the groupoid cardinality of the groupoid B FA ,A (C) of flags in B of type A ,A. This groupoid is discrete so that we have B B |FA ,A (C)| = π0 (FA ,A (C)). But these numbers are precisely the structure constants of Hall(C)op which proves the claim. 36. 35 shows that the 2-fibers of the functor F : X2 → X1 are discrete. This means that we could replace F(A) by the abelian group F(A)Z of finitely supported functions taking values in Z instead of Q.

Sequences of natural numbers with finitely many nonzero terms. Here hα = hα1 hα2 · · · , generalizing the definition for partitions. Finally, we may express the last line by summing over partitions to obtain hα (x)y α = hλ (x)mλ (y) α λ To show (2), we work with variables x1 , . . , xn and y1 , . . ρ (x)y β aρ (x) = β σ where the sum ranges over all β ∈ Nn . We now use the formula aρ det(hβi −n+j ) = aβ from the proof of the Jacobi-Trudi formulas in Lecture 8 which implies that the last line equals aβ (x)y β .

Under this correspondence, we have d(J) = d(α, β). Further, for all k-subspaces N ⊂ S ⊂ M which lie in a fixed Schubert cell CJ , the quotient M/N has the same type λ ∼ β. Therefore, only those Schubert cells so that M/N has type ν contribute to the count and we obtain precisely the claimed formula. 1 Hall algebras via groupoids 2-pullbacks Let C be a category. Consider a diagram /Zo X Y in C and assume that the pullback X ×Z Y exists. From the universal property of pullbacks, we deduce: (A) An isomorphism / X Zo ∼ = / X Y ∼ = Z o ∼ = Y of diagrams in C induces an isomorphism of pullbacks ∼ = X ×Z Y −→ X ×Z Y .