Subrepresentation semirings and an analogue of 6j symbols

Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2007 Subrepresentation semirings and an analogue of 6j- symbols Nam Hee Kwon Louisiana State University and Agricultural and Mechanical College Follow this and additional works at: https://digitalcommons.edu/gradschool_dissertations Part of the Applied Mathematics Commons Recommended Citation Kwon, Nam Hee, "Subrepresentation semirings and an analogue of 6j-symbols" (2007). LSU Doctoral Dissertations.edu/gradschool_dissertations/2720 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please contactgradetd@lsu.

SUBREPRESENTATION SEMIRINGS AND AN ANALOGUE OF 6j-SYMBOLS A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics by Nam Hee Kwon B., Inha University, South Korea, 1996 M., Seoul National University, 1999 May 2007 Acknowledgments It is my great pleasure to express my sincere appreciation to my thesis advisor, Prof. Without his invaluable advice, this dissertation would not be com- pleted. I also would like to express my special thanks to the committee members, Prof. Pramod Achar, Prof.

Jorge Morales, Prof. Gestur Olafsson, Prof. Lawrence Smolinsky and Prof. I am grateful to the Department of Mathematics at Louisiana State University for the financial support during my doctoral program.

Most importantly, I would like to thank my wife, Sunghye Shin, and my children, Yesueng and Yein, for their love and encouragement. I dedicate this dissertation to my parents, Guntaek Kwon and Yiesoon Yang. ii Table of Contents Acknowledgments .1 Two Basic Lemmas .2 Representation Theory of SU (2) .3 The Classical 3j and 6j Symbols .4 Quasi Simply Reducible Groups .2 A Specific Case of End(V ) .3 G-invariant Ideals and Subalgebras. Structure Constants of SSU (2) (End(V )) and The Vanishing of 6j- Symbols .2 Structure Constants of SSU (2) (End(V )).

Subrepresentation Semirings over Quasi Simply Reducible Groups 27 4.1 Twisted Dual and Homomorphism Modules .3 Structure Constants and The Vanishing of Twisted 6j-Symbols. Frobenius-Schur Invariants, Even and Odd Representations .1 New Frobenius-Schur Invariants .2 Even and Odd Representations. Some Properties of Twisted 6j-Symbols .1 Connection with The Classical 3j-Symbols .2 Properties of The Classical 3j-Symbols .4 Properties of The Twisted 6j-Symbols. 66 iii Abstract Let G be a quasi simply reducible group, and let V be a representation of G over the complex numbers C.

In this thesis, we introduce the twisted 6j-symbols over G which have their origin to Wigner’s 6j-symbols over the group SU (2) to study the structure constants of the subrepresentation semiring SG (End(V )), and we study the representation theory of a quasi simply reducible group G laying emphasis on our new G-module objects. We also investigate properties of our twisted 6j-symbols by establishing the link between the twisted 6j-symbols and Wigner’s 3j-symbols over the group G. iv Introduction In this thesis, we examine the structure of the semiring of subrepresentations of certain matrix algebra on which a group acts by algebra automorphism. The study of this semiring, while very natural from a representation-theoretic perspective, was first motivated by a problem in materials science.

We begin by describing how this semiring arises in the theory of composite materials. In materials science, it is an important problem to create a composite material with desired properties. However, it is not easy to predict effective properties of composites because their physical properties are usually strongly dependent on the microstructure. For these reasons, it is natural to consider the set of all pos- sible values of a given physical properties of a composite material that is made with given materials of fixed proportions as one changes the microstructure of the composite.

We call this set a G-closure. It is a subset of an appropriate vec- tor space tensors. Even though most G-closure sets have a non-empty interior, in exceptional cases they degenerate to a surface which is called an exact relation. Because exact relations give the information about a composite material regard- less of its microstructure, it has been an important problem in materials to find such exact relations.

Unfortunately, the classical approach to find exact relations through analytical computations was limited by heavily dependence on the details of the physical context. Moreover, these techniques could not be used to determine whether all exact relations in a specific contest had been found. Recently, in [GMS] the authors developed an abstract theory of exact relations, which not only led to the discovery of many new exact relations, but also gave complete lists in many physical situations. The success of this approach was due 1 to its reduction of the problem of finding exact relations to algebraic questions concerned with the representation theory of the group SO(3).

More specifically, it was shown that finding an exact relation was equivalent to solving an equation involving the multiplication of subrepresentations in a certain matrix SO(3). We now define subrepresentation semiring following [S2]. Let A be an associative algebra with identity over a field k. Assume that the algebra A has a G-module structure with the additional property α·(xy) = (α·x)(α·y) for α ∈ G and x, y ∈ A.

In other words, G acts on A by algebra automorphism. We call A a G-algebra. For a given G-algebra A, let SG (A) be the set of all subrepresentations (i. Then we can give a semiring structure on SG (A) with the usual addition of subspaces and multiplication given by XY = span{xy | x ∈ X, y ∈ Y }.

We call the semiring SG (A) the subrepresentation semiring of the G-algebra A. A fundamental example is given by A = End(V ), where V is a representation of a group G. This was the case that arose in the study of exact relation. In par- ticular, it was shown in [GMS] how the search for exact relations reduce to the algebraic problem of computing the structure constants of certain subrepresenta- tion semiring SG (End(V )) for G = SO(3).

Moreover, it was observed by Etingof and Sage independently that the structure constants of SSO(3) (End(V )) are in fact related to the vanishing of Wigner’s 6j-symbols which arise in the quantum theory of angular momentum. It is well known that there is a double covering homo- morphism π : SU (2) −→ SO(3), and π yields a canonical isomorphism between SSO(3) (End(V )) and SSU (2) (End(V )). Here V denotes a representation of SO(3) over C. Thus for a given representation V over the complex numbers C the struc- ture constants of SSU (2) (End(V )) are also related to the vanishing of Wigner’s 6j-symbols.

This is an unexpected link because Wigner initially developed his 6j- symbols for SU (2) in the quite different context of the quantum theory of angular 2 momentum. Wigner himself generalized his construction of 6j-symbols to a more general class of groups called simply reducible groups, and Sharp further generalized them to quasi simply reducible groups. Quasi simply reducible groups were intro- duced by Mackey in [M3]; their representation theory has broad similarities to the representation theory of the group SU (2). Recall that every irreducible representa- tion of the group SU (2) can be parameterized by the set of nonnegative half integer 1 Z , and each irreducible representation is self-dual (i.

Moreover, the tensor product of two irreducible representations of SU (2) is multiplicity- free, which can be easily checked by the Clebsch-Gordan formula. Keeping these rep- resentation theoretic properties in mind, we define quasi simply reducible groups as follows. A finite or compact group G is called a quasi simply reducible group if there exists an involutory anti-automorphism on G that leaves the conjugacy classes invariant, and irreducible representations of G satisfy the multiplicity-free property. If we take the involutory anti-automorphism on G to be the multiplica- tion inverse, then in this case we call the group G a simply reducible group.

SU (2) is the fundamental example of a simply reducible group. Now the following natural question arises: How are the structure constants of SG (End(V )) related to the 6j-symbols over G when we replace the group SU (2) by an arbitrary quasi simply reducible group G? To answer this question, in this thesis we explicitly calculate the structure con- stants of the subrepresentation semiring SG (End(V )) by introducing a new class of 6j-symbols over G which we will call the twisted 6j-symbols in this thesis. Before giving a more detailed description of the content of this thesis, we briefly digress to mention that the subrepresentation semiring SG (End(V )) can also be used to describe the G-invariant ideals and subalgebras of End(V ). 3 Let A be a G-algebra, and let I be a G-invariant left ideal of A.

Then we define the saturation of I by I = {J ∈ SG (A) | J ⊂ I}. Clearly I is a saturated left ideal containing the maximum element I, where we consider the inclusion as a partial 0 0 order on SG (A) (Saturated means that if J ∈ I and J ⊂ J, then J ∈ I). Thus we can assign each G-invariant left ideal I of A to the saturated left ideal I of SG (A) containing a maximum element, and this mapping is a bijective correspon- dence. Furthermore, when A = End(V ) it is known explicitly about the types of saturated left ideals of SG (End(V )).

More precisely, let W be a subrepresentation of V. Then we define G-invariant left ideal Ann(V ) called annihilator of W by Ann(W ) = {f ∈ End(V ) | f (W ) = 0}. It is known that every saturated left ideal of SG (End(V )) is of the form Ann(W ) (see [S1]). Similarly, for a given G-algebra A there is a bijection between G-invariant subalgebras and saturated subhemirings of SG (A) containing their maximum elements.

Recall that we call a set R a hemiring if R is an additive monoid under multiplication, but not containing the unity. In the G-algebra End(V ), it is also known that every nonzero saturated subhemiring is given by the saturation of a certain induced G-module. A complete description of the invariant subalgebras is given in [S2] for the case of V irreducible. Now we give an in depth description of the contents of this thesis.

From Chapter 1 to 3, we cover basic material and motivations for this thesis. The main results are exhibited in Chapter 4 through Chapter 6. In Chapter 1, we review the classical 3j and 6j symbols through the represen- tation theory of the Lie algebra sl(2, C). Our approach is different from that of Wigner.

It does not generalized to arbitrary groups, but it is quickened and more elegant for SU (2). We also introduce the definition of quasi simply reducible groups and give some examples of quasi simply reducible groups. 4 In Chapter 2 and 3, we give some background on subrepresentation semirings. In Chapter 2, we define subrepresentation semiring and recall some basic concepts.

We then focus on the important class of subrepresentation semirings coming from central simple algebras of the form End(V ), where V is a representation. In Chap- ter 3 we review how the structure constants of SSU (2) (End(V )) are related to the vanishing of 6j-symbols. In Chapter 4, we consider a group G endowed with an involutory anti-automorphism. we first introduce new G-modules which are called twisted dual G-modules and twisted homomorphism G-modules respectively.

As vector spaces, these will coin- cide the usual notion of dual spaces V ∗ and homomorphism spaces Hom(V, W ), but they will have new G-module structures. Using these new G-modules, we define Clebsch-Gordan coefficients and twisted 6j-symbols for a quasi simply reducible group G. Then in Theorem (4.