We present a general diagrammatic approach to the construction of efficient algorithms for computing the Fourier transform of a function on a finite group. By extending work which connects Bratteli diagrams to the construction of Fast Fourier Transform algorithms we make explicit use of the path algebra connection to the construction of Gel'fand-Tsetlin bases and work in the setting of quivers. We relate this framework to the construction of a configuration space derived from a Bratteli diagram. In this setting the complexity of an algorithm for computing a Fourier transform reduces to the calculation of the dimension of the associated configuration space. Our methods give improved upper bounds for computing the Fourier transform for the general linear groups over finite fields, the classical Weyl groups, and homogeneous spaces of finite groups, while also recovering the best known algorithms for the symmetric group and compact Lie groups.Date: December 9, 2015. 2000 Mathematics Subject Classification. To be filled in.Definition 2.7. A quiver Q is a directed multigraph with vertex set V (Q) and edge set E(Q). For an arrow (directed edge) e ∈ E(Q) from vertex β to vertex α, we call α the target of e and β the source of e.Let Q be a quiver. For each e ∈ E(Q), let t(e) denote the target of e and s(e) the source of e. Definition 2.8. A quiver Q is graded if there is a function gr : V (Q) → N such that for each e ∈ E(Q), gr(t(e)) > gr(s(e)).Example 2.9. Figure 1 is an example of a graded quiver. Each vertex v is labeled by its grading, gr(v).

Abstract. We present a general diagrammatic approach to the construction of efficient algorithms for computing a Fourier transform on a semisimple algebra. This extends previous work wherein we derive best estimates for the computation of a Fourier transform for a large class of finite groups. We continue to find efficiencies by exploiting a connection between Bratteli diagrams and the derived path algebra and construction of Gel'fand-Tsetlin bases. Particular results include highly efficient algorithms for the Brauer, Temperley-Lieb algebras, and Birman-Murakami-Wenzl algebras.

Abstract. We specify a class of graphs, H t , and characterize the irreducible decompositions of all powers of the cover ideals. This gives insight into the structure and stabilization of the corresponding associated primes; specifically, providing an answer to the question "For each integer t ≥ 0, does there exist a (hyper) graph H t such that stabilization of associated primes occurs at n ≥ (χ(H t ) − 1) + t?" [4]. For each t, H t has chromatic number 3 and associated primes that stabilize at n = 2 + t.

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International audience We present a general diagrammatic approach to the construction of efficient algorithms for computingthe Fourier transform of a function on a finite group. By extending work which connects Bratteli diagrams to theconstruction of Fast Fourier Transform algorithms we make explicit use of the path algebra connection and work inthe setting of quivers. In this setting the complexity of an algorithm for computing a Fourier transform reduces to pathcounting in the Bratelli diagram, and we generalize Stanley's work on differential posets to provide such counts. Ourmethods give improved upper bounds for computing the Fourier transform for the general linear groups over finitefields, the classical Weyl groups, and homogeneous spaces of finite groups.