On My Schur-Weyl Duality Project Aba Mbirika Department of Mathematics University of Wisconsin–Eau Claire Eau Claire, WI 54701 Email: Mbirika@Uwec.Edu

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On my Schur-Weyl duality project Aba Mbirika Department of Mathematics University of Wisconsin–Eau Claire Eau Claire, WI 54701 email: [email protected]

1. Brief introduction to the project A focal point of classical representation theory is Schur-Weyl duality. Let V = Cn. Both ⊗k the general linear group GLn(C) and the symmetric group Sk act on V , the k-fold tensor product of V . In 1927, Schur found that these two groups are mutual centralizers of each other, thus relating their representations [4]. Subgroups of GLn and their centralizers have been extensively studied. We list a few with their corresponding centralizers below each one.

GLn(C) ⊇ On(C) ⊇ Sn ⊇ An CSk ⊆ CBk(n) ⊆ CPk(n) ⊆ Unknown

The centralizer of GLn can be viewed as the span of permutation diagrams. The centralizer of the orthogonal group On is the Brauer algebra CBk(n), spanned by the permutation diagrams plus some extra diagrams [1]. In 1993, the centralizer of the subgroup Sn of permutation matrices sitting inside GLn was presented as the partition algebra CPk(n), a span of partition diagrams [2]. The alternating group An is the determinant 1 permutation matrices. Its centralizer has yet to be formulated. It should be a super-algebra of this so-called partition algebra. I am attempting to describe this centralizer.

2. Background and motivation n Let V = C . The group GLn(C) acts naturally on V by left multiplication. Consider ⊗k ⊗k k-fold tensor copies of V , denoted as V . We may extend the action of GLn(C) on V by

g · (v1 ⊗ v2 ⊗ · · · ⊗ vk) = (g · v1) ⊗ (g · v2) ⊗ · · · ⊗ (g · vk). ⊗k Also, the symmetric group Sk acts naturally on V by permuting the tensor places, that is, σ · (v1 ⊗ v2 ⊗ · · · ⊗ vk) = vσ(1) ⊗ vσ(2) ⊗ · · · ⊗ vσ(k). Schur’s famed result revealed that GLn(C) and Sk are mutual centralizers of each other [4]. We view Sk as permutation diagrams—that is, 2-row diagrams with k vertices in each row and a one-to-one correspondence of these two sets of vertices given by edges connecting them. Thus, every vertex has degree one and there are k edges. We may view Sn as sitting inside GLn(C) as the set of permutation matrices so that V is the permutation representation. Restricting the action of GLn(C) to this subgroup, we ⊗k get an action of Sn on V . Independently, Vaughn Jones in 1993 and Paul Martin in 1991 investigated the centralizer of this action. They deduced that this centralizer corresponds to the span of a set of diagrams which we call the partition algebra. As a program resident at MSRI in 2008, I had many discussions with Tom Halverson about investigating the centralizer of the alternating group An viewed as sitting inside GLn(C) as the determinant 1 permutation matrices. The centralizer of this subgroup acting on V ⊗k should be a super-algebra of the partition algebra. 1 3. My results and future work

A ﬁrst step in this direction is to deduce how irreducible representations of An decompose when we restrict or induce them.

• Project 1. (Completed) Deduce the branching rules for An. To accomplish this, I used the known branching rules of the symmetric group, some Cliﬀord and Mackey theory, and Frobenius reciprocity. Details of the proof can be found in my draft [3]. It is known from Jones’s work [2] on the partition algebra, that a map is in the centralizer of Sn, if and only if, its matrix entries are constant on the Sn-orbits of its matrix coordinates. The orbits decompose a set of size 2k into subsets and thus correspond to set partitions. Applying this same reasoning to the alternating group analogue, I performed a number of calculations using GAP and Sage software.

• Project 2. (Completed) For n = 4 and k = 2, I computed the S4-orbits on V ⊗ V by letting S4 act on the 256 matrix entries of a 16 by 16 matrix. As expected there were 15 orbits corresponding to the 15 set partitions of the numbers 1 through 4. In the alternating group analogue, raw computer data gave us 7 additional maps when looking at the A4-orbits. These small calculations took tremendous amounts of computer time and memory. However, they give us the ﬁrst ever glimpse of these centralizer maps. The raw computer data essentially gave us the material to describe exactly the maps that centralize the action of A4 on a basis vector in V ⊗ V . However, we do not yet know how these maps multiply. We have a conjecture on exactly which Sn-orbits split in the An case, thus predicting the additional 7 maps when n = 4. • Project 3. (Future Work – ideal for undergraduate research) The raw data computed above gave us an idea of what the new diagrams might look like. However to construct an algebra, a multiplication on these new diagrams must make sense. Projects involving the construction of this diagram algebra would be suitable for undergraduate research. References [1] R. Brauer, On algebras which are connected with the semisimple continuous groups, Ann. Math. 38 (1037), 854–872. [2] V. Jones, The Potts model and the symmetric group, Subfactors: Proceedings of the Taniguchi Symposium on Operator Algebras (1993), 259–267. [3] A. Mbirika, Towards a Schur-Weyl duality for the alternating group, Draft in progress (2009). (Available at http://people.uwec.edu/mbirika/MSRI-result-Summer2009_draft.pdf). [4] I. Schur, Uber¨ die rationalen Darstellungen der allgemeinen linearen Gruppe, (1927). Reprinted in I. Schur, Gesammelte Abhandlungen III (1973), 68–85.