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Gems from the Work of Georgia Benkart

Tom Halverson
Arun Ram

Communicated by Notices Associate Editor Steven Sam

Graphic without alt text

Georgia Benkart completed her PhD in 1974 at Yale University, where she was the 30th of Nathan Jacobson’s 34 PhD students. From there she joined the faculty at the University of Wisconsin–Madison, where she is now Professor Emerita. Since her retirement from teaching she has provided tremendous service to the mathematical community, notably as President of the AWM and as an Associate Secretary of the AMS for more than a decade.

In this article we highlight a few selected gems from her extensive contribution to our field, organized in a roughly chronological sequence of vignettes and images (which can be read or viewed in any order). Our hope is that we can capture and transmit a snapshot of Georgia’s rich mathematics, beautiful style, and wonderful mathematical personality.

Classifying simple Lie algebras

In algebra in 1974, the air was thick with the classification of finite simple groups, with new finite simple groups being discovered in a frenzy, and the question always in the air:

“Have we found them all?”

At that time there was another such classification effort beginning: a search for all of the finite-dimensional simple Lie algebras.

In characteristic 0 the problem had been completed by Cartan and Killing around 1894, resulting in the list of Dynkin diagrams (Figure 1), which are in bijection with the finite-dimensional simple Lie algebras. Over an algebraically closed field of characteristic , four additional series occur:

the Witt Lie algebras ,

the special Lie algebras ,

the Hamiltonian Lie algebras ,

the contact Lie algebras .

The monograph by Benkart, Gregory, and Premet BGP09 provides complete details on these algebras. They are known as the generalized Cartan-type Lie algebras, because they are derived from Cartan’s four infinite families (Witt, special, Hamiltonian, contact) of infinite-dimensional complex Lie algebras. Cartan’s work set the stage for Kostrikin–Šafarevič KŠ66, who identified the above four unifying families of simple Lie algebras living in the Witt algebras. Earlier work of George Seligman Sel67 (also at Yale) emphasized the role and the importance of the Lie algebras of Cartan type. George was one of Jacobson’s first students and Georgia was one of his last.

In 1966, Kostrikin and Šafarevič conjectured that the Cartan-type Lie algebras and the Lie algebras coming from characteristic 0 were all of the finite-dimensional simple Lie algebras (over an algebraically closed field) in characteristic . The original formulation was for “restricted” Lie algebras, and the general statement for finite-dimensional simple Lie algebras is the “Generalized Kostrikin–Šafarevič conjecture.”

Figure 1.

The classification of finite-dimensional simple Lie algebras in characteristic 0 is by the above Dynkin diagrams. In characteristic there are five additional series of algebras: (1) the Witt Lie algebras ,(2) the special Lie algebras , (3) the Hamiltonian Lie algebras , and (4) the contact Lie algebras ; and when there is one more additional series: (5) the Melikyan Lie algebras .

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The study and proof of the Kostrikin–Šafarevič conjecture inspired work by many people around the world. Georgia brought to Wisconsin the mindset of the Jacobson school, emphasizing a module-theoretic approach to the classification of algebraic systems. She joined a thriving algebra community that included Marty Isaacs, Marshall Osborn, Donald Passman, and Louis Solomon. Also in the thick of the action around the Kostrikin–Šafarevič conjecture were Richard Block, Robert Wilson (who had been a student of George Seligman at Yale), and Victor Kac, who seemed to be everywhere, classifying all things Lie.

Benkart and Osborn BO84 classified the finite-dimensional simple Lie algebras of characteristic with a one-dimensional Cartan subalgebra, showing that they are either or Albert–Zassenhaus Lie algebras (the algebras and a family of Hamiltonian Lie algebras). Their paper BO90 studied the subalgebra of a finite-dimensional simple Lie algebra determined by a root . Modulo the radical, these one-sections are isomorphic to either , , or to a subalgebra of containing .

The results of Benkart and Osborn, along with their proof techniques, were ultimately absorbed into the general classification process. In the 1990s, Alexander Premet and Helmut Strade pulled it all together, methodically completing every step to a full classification.

Of course, as with any huge project, there were many other important contributors in addition to those named here. In the middle of it all, in 1980, Melikyan found a new finite-dimensional simple Lie algebra in characteristic 5, of dimension 125. That certainly put a wrench into things, and increased the worry that, in those small cases, there might exist even more fascinating and untamed algebras that nobody had seen before. Fortunately, now the whole project is finished for and is comprehensively exposited in the 1100 pages of the three volumes of Helmut Strade’s books, Simple Lie algebras over fields of positive characteristic Vols. I, II, and III Str17aStr17bStr13.

Quoting from the Math Review of Vol. III:

Kac’s recognition theorem is one major result whose proof is not included in the book. All details for an arbitrary can be found in a paper of G.M. Benkart, T.B. Gregory and Premet BGP09.

The Recognition Theorem was a hugely important step on the long road to completion of the classification. To quote from the introduction of BGP09: “The Recognition Theorem is used several times throughout the classification; its first application results in a complete list of the simple Lie algebras of absolute toral rank two, and its last application yields a crucial characterization of the Melikyan Lie algebras, thereby completing the classification.” Finally those mysterious Melikyan algebras (they had multiplied in the interim and become a whole family) were under control in the sense that the freedom that causes them to appear had been pinpointed, and it had been checked carefully that this freedom doesn’t cause other sporadic examples of this nature. The monograph of Benkart, Gregory, and Premet is a wonderful work to read: thorough, efficient, elementary, with precise definitions; it contains a clear big-picture point of view. It is absolutely beautifully written.

Infinite dimensions and magic squares

Figure 2.

Georgia Benkart on February 22, 1979 during a visit to Indiana University. An image from the Paul R. Halmos Photograph Collection.

Graphic for Figure 2.  without alt text

The structure of a finite-dimensional Lie algebra corresponding to one of the Dynkin diagrams in Figure 1 is governed by its root system , and decomposes into a direct sum of the form,

and equal to the number of vertices in the Dynkin diagram. Furthermore, the root system has a geometric description connecting it to the world of polytopes (see Figure 3).

Figure 3.

The root system for a Lie algebra corresponding to the Dynkin diagram . The root system consists of the vectors from the center to the vertices and from the center to the midpoints of the edges of the octahedron. See equation 1.

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The second half of the 20th century produced a huge expansion into the universe of infinite-dimensional Lie algebras. The finite-dimensional Cartan-type Lie algebras in Figure 1 are the characteristic versions of infinite-dimensional characteristic 0 Lie algebras that arose from Cartan’s study of “pseudogroups.” The study of Feynman path integrals and the development of string theory also produced new examples of infinite-dimensional Lie algebras with interesting structure.

The underlying structure of the infinite-dimensional Lie algebra comes from a finite-dimensional sitting inside . This property was formalized in the early 1990s by Berman and Moody when they defined -graded Lie algebras. A -graded Lie algebra contains a subalgebra corresponding to a Dynkin diagram, and the whole Lie algebra decomposes into root spaces indexed by the root system of ,

Berman and Moody classified the -graded Lie algebras for which the Dynkin diagram does not have double or triple edges by viewing them as Lie algebras analogous to , where is an (associative) algebra. Favorite examples are the polynomial rings and the Laurent polynomial rings , but can be much more general.

Berman and Moody’s classification leads one to wonder what happens when the Dynkin diagram has multiple edges. Efim Zelmanov had started to study these cases, and in the course of his work gave a few lectures in the seminar at the University of Wisconsin. One morning Georgia came in and indicated that she thought that some of the ideas from her thesis might apply to this question. It didn’t take long before Georgia and Efim hunkered down and quickly polished off all the other cases and completed the amazing theorem that

where is a finite-dimensional Lie algebra with root system , is a small -module, and is a subalgebra of derivations that acts on the algebra . Hence the infinite-dimensional Lie algebra is something like that in Figure 4, where and are visualized as appendages to the root system . The Benkart–Zelmanov paper BZ96 explaining how this works has become a classic.

Georgia didn’t stop there. There are two basic steps in the classification of -graded Lie algebras:

First: One has to show that the only possible forms that a -graded Lie algebra can take are .

Second: After narrowing down the possibilities, one has to show that they all occur in reality and do, in fact, produce -graded Lie algebras.

This second step is obtained by powerful constructions which go by various names (see Tables 1 and 2): “Freudenthal’s magic square,” the “Tits–Kantor–Koecher construction,” “generalized octonions.” These constructions were originally conceived to build the Lie algebras corresponding to the Dynkin diagrams and . They were vastly generalized by Benkart–Zelmanov to construct -graded Lie algebras and by Benkart–Elduque and Elduque to extend to exceptional Lie superalgebras and Lie algebras and Lie superalgebras in characteristic .

Figure 4.

The infinite-dimensional -graded Lie algebra corresponding to the root system for the Dynkin diagram . The octahedron provides the structure of the root system of the finite-dimensional Lie algebra . The -graded Lie algebra is built by fitting -modules and into sockets on the mother board labeled by the elements of . See 2.

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Table 1.

The Tits–Kantor–Koecher construction. In this table , denotes the algebra of matrices with entries from a field , and is the Jordan algebra of Hermitian matrices over the unital composition algebra .

Table 2.

Freudenthal’s magic square or the symmetric (Vinberg) construction. The rows are indexed by , and the columns are indexed by .

The construction has two forms: the first method is to take a Jordan algebra and a composition algebra and twist them together to get a Lie algebra . The other version of the construction (introduced by Vinberg) builds the Lie algebra from two composition algebras and . In this version, the symmetry of Freudenthal’s magic square is embedded into the construction.

The wonderful article of Elduque in the Tits 80th birthday volume Eld11 provides an accessible survey of the various constructions of Freudenthal’s magic square, along with recent advances in the theory involving Georgia and her coauthors and a nice entrée into open questions and current research in this vein. The original paper of Benkart–Zelmanov BZ96 classified -graded Lie algebras for the cases where the Dynkin diagram of is , , , and . The AMS Memoir of Allison, Benkart, and Gao ABG02 provides an amazing resource for understanding all parts of the classification of -graded Lie algebras, the analysis of their derivations, central extensions and invariant forms, and their constructions, including the Tits–Kantor–Koecher constructions.

Elemental Lie algebras

Imagine that it is the early 1800s and you are Dalton, or Gay-Lussac, or Avogadro, trying to figure out how atoms combine to make molecules. There are two fundamental problems to solve:

(a)

What are the individual elements?

(b)

How do they combine to make molecules?

Now imagine that it is the turn of the 21st century and you are Georgia Benkart trying to figure out how Lie algebras are built. There are two fundamental problems:

(a)

What are the littlest Lie algebras?

(b)

How do they combine to make larger Lie algebras?

A motivating phenomenon is that all finite-dimensional simple Lie algebras (in characteristic ) and all Kac–Moody Lie algebras are constructed from the little Lie algebras glued together appropriately.

Letting ,

where

and

Another little Lie algebra is the three-dimensional Heisenberg Lie algebra

where

and

These algebras are strikingly similar in presentation, but different in application. If one has these examples in mind, then it is not very surprising that Georgia has sequences of papers engaged in the study of families of “elemental” algebras over a field  :

(A) the parametric family

(B) the down-up algebras BR98, depending on parameters :

If , , and , then , and we recover the Heisenberg algebra. This is because in , the relation expands to .

Figure 5.

Irreducible modules for the down-up algebras . Up to constants depending on the parameters , the operators act according the red edges and the operators act according the blue edges. The black vertex represents the highest weight and the lowest weight, respectively. See 4.

\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{SVG} $$ \begin{array}{ccc} \begin{tikzpicture} \node[norep] (HW1) at (-1.5,1) {}; \node[rep] (HW2) at (-1.5,2) {}; \node[rep] (HW3) at (-1.5,3) {}; \node[hiweight] (HW4) at (-1.5,4) {}; \node(botdots) at (-1.5,.9) {\LARGE$\vdots$}; \draw(HW1) edge[up,out=60, in=-60,right] node {${u}$} (HW2); \draw(HW2) edge[down,out=-120, in=120,left] node {${d}$} (HW1); \draw(HW2) edge[up,out=60, in=-60,right] node {${u}$} (HW3); \draw(HW3) edge[down,out=-120, in=120,left] node {${d}$} (HW2); \draw(HW3) edge[up,out=60, in=-60,right] node {${u}$} (HW4); \draw(HW4) edge[down,out=-120, in=120,left] node {${d}$} (HW3); \end{tikzpicture} &\begin{tikzpicture} \node[hiweight] (LW1) at (0,1) {}; \node[rep] (LW2) at (0,2) {}; \node[rep] (LW3) at (0,3) {}; \node[norep] (LW4) at (0,4) {}; \node(topdots) at (0,4.5) {\LARGE$\vdots$}; \draw(LW1) edge[up,out=60, in=-60,right] node {${u}$} (LW2); \draw(LW2) edge[down,out=-120, in=120,left] node {${d}$} (LW1); \draw(LW2) edge[up,out=60, in=-60,right] node {${u}$} (LW3); \draw(LW3) edge[down,out=-120, in=120,left] node {${d}$} (LW2); \draw(LW3) edge[up,out=60, in=-60,right] node {${u}$} (LW4); \draw(LW4) edge[down,out=-120, in=120,left] node {${d}$} (LW3); \end{tikzpicture} & \begin{tikzpicture} \node[norep] (DI1) at (1.5,1) {}; \node[rep] (DI2) at (1.5,2) {}; \node[rep] (DI3) at (1.5,3) {}; \node[norep] (DI4) at (1.5,4) {}; \node(topdots) at (1.5,4.5) {\LARGE$\vdots$}; \node(botdots) at (1.5,.9) {\LARGE$\vdots$}; \draw(DI1) edge[up,out=60, in=-60,right] node {${u}$} (DI2); \draw(DI2) edge[down,out=-120, in=120,left] node {${d}$} (DI1); \draw(DI2) edge[up,out=60, in=-60,right] node {${u}$} (DI3); \draw(DI3) edge[down,out=-120, in=120,left] node {${d}$} (DI2); \draw(DI3) edge[up,out=60, in=-60,right] node {${u}$} (DI4); \draw(DI4) edge[down,out=-120, in=120,left] node {${d}$} (DI3); \end{tikzpicture} \\ \text{highest weight} &\text{lowest weight} &\text{doubly infinite} \end{array} $$ $$ \begin{array}{c} \begin{tikzpicture} \node[rep] (FD1) at (5., 2.5) {}; \node[rep] (FD2) at (4.5, 3.36603) {}; \node[rep] (FD3) at (3.5, 3.36603) {}; \node[rep] (FD4) at (3., 2.5) {}; \node[rep] (FD5) at (3.5, 1.63397) {}; \node[rep] (FD6) at (4.5, 1.63397) {}; \par\draw(FD5) edge[up,above,out=50,in=140] node {${u}$} (FD6); \draw(FD6) edge[down,below,out=210,in=-20,looseness=1] node {${u}$} (FD5); \par\draw(FD6) edge[up,left,out=80,in=230] node {${u}$} (FD1); \draw(FD1) edge[down,right,out=-90,in=20,looseness=1] node {${d}$} (FD6); \par\draw(FD4) edge[up,right,out=-40,in=100] node {${u}$} (FD5); \draw(FD5) edge[down,left,out=160,in=270,looseness=1] node {${d}$} (FD4); \par\draw(FD3) edge[up,right,out=-100,in=40] node {${u}$} (FD4); \draw(FD4) edge[down,left,looseness=1,out=100,in=210] node {${d}$} (FD3); \par\draw(FD2) edge[up,below,out=210,in=-20] node {${u}$} (FD3); \draw(FD3) edge[down,above,looseness=1,out=40,in=150] node {${d}$} (FD2); \par\draw(FD1) edge[up,left,out=140,in=-70] node {${u}$} (FD2); \draw(FD2) edge[down,right,looseness=1,out=-20, in=90] node {${d}$} (FD1); \end{tikzpicture} \\ \text{finite-dimensional} \end{array} $$ \end{SVG}

These algebras and capture the core underlying structures that join together to make larger Lie algebras and their quantum groups. Georgia and her collaborators have done thorough studies of the properties of these “little quantum groups” by determining all of the following: automorphisms, inner automorphisms, centers, derivations, inner derivations, their Hochschild cohomology , prime ideals, primitive ideals, Duflo correspondences between primitive ideals and annihilators of simple modules, highest weight modules, lowest weight modules, finite-dimensional modules, Whittaker modules, and also some tensor product rules for simple modules in case that wasn’t enough already.

Just to highlight a tiny portion of these results, Georgia and her collaborators determine precisely all the possible “shapes” of irreducible modules of down-up algebras . These are shown pictorially in Figure 5.

Because the algebras and are so “elemental” (generalizing the structures from and three-dimensional Hesenberg algebras), one has confidence that they will be useful to mathematicians of the future in the same way that intimate knowledge of Mendeleev’s periodic table is indispensible for any post-19th century chemist. The elemental Lie algebras are the atoms from which larger Lie algebras and quantum groups that arise in nature (i.e., many other parts of mathematics and physics) are built.

Talking the talk: A Tale of Two Groups

In a Dickensian plenary address at the 1994 Joint Math Meetings, Georgia told the story of Schur–Weyl duality as a “Tale of Two Groups.” See Ben96. The protagonist is a group acting on tensor powers of a defining representation, and the antagonist is the algebra of endomorphisms that commute with . See Figure 6.

Figure 6.

Schur–Weyl duality between the general linear group and the symmetric group , between the orthogonal group and the Brauer algebra , and between the symmetric group and the partition algebra .

\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{SVG} $$ \begin{matrix} \begin{tikzpicture}[xscale=.8,yscale=.55] \node(module) at (0,6) {$V^{\otimes n}$} edge [in=200,out=160,loop] node[left] {$G$} () edge [in=340,out=20,loop] node[right] {$\End_G(V^{\otimes n})$} (); \node(dim) at (0,5) {$\scriptstyle{ \dim(V)=r}$}; \node(G1) at (-2,4) {$GL_r(\CC)$}; \node(G2) at (-2,2) {$O_r(\CC)$}; \node(G3) at (-2,0) {$S_r$}; \node(sup) at (-2,3) {$\rotatebox[origin=c]{90}{$\subseteq$}$}; \node(sup) at (2,3) {$\rotatebox[origin=c]{-90}{$\subseteq$}$}; \node(sup) at (-2,1) {$\rotatebox[origin=c]{90}{$\subseteq$}$}; \node(sup) at (2,1) {$\rotatebox[origin=c]{-90}{$\subseteq$}$}; \node(C1) at (2,4) {$\CC S_n$}; \node(C2) at (2,2) {$B_n(r)$}; \node(C3) at (2,0) {$P_n (r)$}; \draw[dotted] (G1) to (C1); \draw[dotted] (G2) to (C2); \draw[dotted] (G3) to (C3); \end{tikzpicture} \end{matrix} $$ \end{SVG}

In his groundbreaking thesis at the turn of the 20th century, Schur used these methods to construct the irreducible polynomial representations of the general linear group . He showed that is generated by , the algebra of permutations, displayed here as a permutation diagram,

In the 1930s Brauer showed that if , then is generated by the algebra of Brauer diagrams, which correspond to arbitrary matchings of vertices,

In about 1990, Paul Martin and Vaughan Jones showed that if , the symmetric group, then the centralizer is generated by set partition diagrams,

Set partition diagrams multiply with one another via concatenation:

Schur–Weyl duality allows information to flow back and forth between the group and its centralizer . In an AMS Memoir BBL90 Georgia and her coauthors, Dan Britten and Frank Lemire, study finite-dimensional representations of , , , and . They identify submodules for these inside the tensor space and use the combinatorics of the centralizer, for example,

to understand stability properties for irreducible -modules as grows. Georgia Ben90 and Sheila Sundaram Sun90 each give elegant descriptions of these combinatorial methods in representation theory.

In 1989, in a collaboration BCH94 with five graduate students at the University of Wisconsin, Georgia defined the walled-Brauer algebra by determining the centralizer of the on , where is the dual module to . This time, the diagrams come with a left part and a right part separated by a wall, with the constraint that horizontal edges must cross the wall and top-to-bottom edges must not cross the wall,

This collaboration, with Georgia leading a group of five junior mathematicians at once, was unusual at the time. Now, this is more common and one finds, among Georgia’s recent papers, several team collaborations that include early-career researchers who have been stimulated by Georgia’s leadership. Not only is Georgia a natural and inspiring mentor for these teams, but she initiated them long before there were organizations like Banff (see Figure 11) and MSRI helping so effectively to make it happen.

Walking the walk: The Representation Theory Way

In 2014, Georgia delivered the Noether Lecture at the International Congress of Mathematicians in Seoul, Korea entitled, “Walking on Graphs the Representation Theory Way.” The motivating idea is that one can build

by applying idempotents of to . The idempotent

is a projection onto the irreducible summand . A powerful way to study this is by building a graph that keeps track of what happens when one tensors by . This representation graph, or McKay quiver, has vertices and edges if appears times in . For example, if is the cyclic group of order , and is the two-dimensional representation of corresponding to the matrix

then the representation graph of the pair is

If is the binary dihedral group of order and is the two-dimensional representation given by the matrices

then the representation graph of the pair is

Finally, if is one of the three polyhedral groups,

and is the two-dimensional representation of , then the representation graphs of the pairs are the graphs in Figure 7. The observation that the graphs , , , , are exactly the “simply-laced affine Dynkin diagrams” is the amazing McKay correspondence. These same graphs also describe (see Kac90 and Bri71) the internal structure of the Lie algebras of loop groups as well as the structure of the subregular nilpotent orbits for reductive algebraic groups!

In these examples, if we now let , then the two commuting actions of and on

give a decomposition of into irreducible -bimodules,

The walks on the representation graph encode multiplicities and dimensions:

and is the number of walks that come back home (to the node labeled ) after steps:

Figure 7.

The representation graphs of the binary tetrahedral, octahedral, and icosahedral groups are the simply-laced affine Dynkin diagrams of type and .

\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{SVG} $$ \begin{array}{lcl} \hat E_6 && \begin{matrix}\begin{tikzpicture}[every node/.style={inner sep=1}, scale=.9, line width=0.7] \node[wV, label=above:{\small$0$}] (0) at (0,0) {}; \node[wV, label=above:{\small$1$}] (1) at (1,0) {}; \node[wV, label=above:{\small$2$}] (2) at (2,0) {}; \node[wV, label=above:{\small$3$}] (3) at (3,0) {}; \node[wV, label=above:{\small$4$}] (4) at (4,0) {}; \node[wV, label=right:{\small$3'$}] (3p) at (2,-1) {}; \node[wV, label=right:{\small$4'$}] (4p) at (2,-2) {}; \draw[line width=0.7] (0) to (1) (1) to (2) (2) to (3) (3) to (4) (2) to (3p) (3p) to (4p); \end{tikzpicture} \end{matrix} \\ \\ \hat E_7 && \begin{matrix} \begin{tikzpicture}[every node/.style={inner sep=1}, scale=.9, line width=0.7] \node[wV, label=above:{\small$0$}] (0) at (0,0) {}; \node[wV, label=above:{\small$1$}] (1) at (1,0) {}; \node[wV, label=above:{\small$2$}] (2) at (2,0) {}; \node[wV, label=above:{\small$3$}] (3) at (3,0) {}; \node[wV, label=above:{\small$4$}] (4) at (4,0) {}; \node[wV, label=above:{\small$5$}] (5) at (5,0) {}; \node[wV, label=above:{\small$6$}] (6) at (6,0) {}; \node[wV, label=right:{\small$4'$}] (4p) at (3,-1) {}; \draw[line width=0.7] (0) to (1) (1) to (2) (2) to (3) (3) to (4) (4) to (5) (5) to (6) (3) to (4p); \end{tikzpicture} \end{matrix} \\ \\ \hat E_8 && \begin{matrix} \begin{tikzpicture}[every node/.style={inner sep=1}, scale=.9, line width=0.7] \node[wV, label=above:{\small$0$}] (0) at (0,0) {}; \node[wV, label=above:{\small$1$}] (1) at (1,0) {}; \node[wV, label=above:{\small$2$}] (2) at (2,0) {}; \node[wV, label=above:{\small$3$}] (3) at (3,0) {}; \node[wV, label=above:{\small$4$}] (4) at (4,0) {}; \node[wV, label=above:{\small$5$}] (5) at (5,0) {}; \node[wV, label=above:{\small$6$}] (6) at (6,0) {}; \node[wV, label=above:{\small$7$}] (7) at (7,0) {}; \node[wV, label=right:{\small$6'$}] (6p) at (5,-1) {}; \draw[line width=0.7] (0) to (1) (1) to (2) (2) to (3) (3) to (4) (4) to (5) (5) to (6) (6) to (7) (5) to (6p); \end{tikzpicture} \end{matrix} \end{array} $$ \end{SVG}

A particularly elegant way to enumerate walks on the representation graph is to expand them into paths on the corresponding Bratteli diagram , which is an infinite lattice organized so that the nodes on level are those that can be reached by an -step walk starting at the root on (see Figure 8).

With several collaborators, Georgia has used walks on these representation graphs to answer many questions in combinatorial representation theory. To name just a few: they describe the projection operators in McKay and Motzkin centralizer algebras; they characterize the kernel of the partition algebra on tensor space; they describe walks on hypercubes; and they are used to perform chip firing on Dynkin diagrams and McKay quivers.

Fusion rules!

Georgia’s most recent talks and collaborations have centered around fusion rules. Fusion matrices encode the rules that determine the decomposition of the tensor product of two modules into a direct sum of simple modules. In the case of the McKay correspondence, the fusion matrices are the adjacency matrices of the representation graphs in Figure 7, and in conformal field theory in physics, integrable models are described by the fusion rules for their charges.

In a group project BBK21 that began at the workshop in Leeds for Women in Noncommutative Algebra and Representation Theory (WINART3), Georgia and her collaborators compute fusion matrices for certain classes of finite-dimensional Hopf algebras. They express the eigenvalues and eigenvectors of these matrices in terms of Chebyshev polynomials, furthering the case that Chebyshev polynomials are as dense in representation theory as they are in numerical analysis. A key step is to relate the eigenvectors to characters, and an overarching question in this work is to find a good notion of a character table for a Hopf algebra.

Figure 8.

The Bratteli diagram for . Surprising and beautiful things happen in this diagram. The Dynkin diagram is embedded at the top of the Bratteli diagram (shaded in blue). The dimension of the irreducible -modules are the red labels, which satisfy a Pascal’s triangle-like addition rule. The dimension is the number of paths ending at 0 on level , i.e., the numbers . Thus the red number at node 0 on level is the sum of the squares of the red numbers on level . For example, .

\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{SVG} $$ \begin{matrix} \begin{tikzpicture}[every node/.style={inner sep=1}, scale=.9, line width=0.7] \node(k0) at (-1,0) {$\scriptstyle{n=0:}$}; \node(k1) at (-1,-1) {$\scriptstyle{n=1:}$}; \node(k2) at (-1,-2) {$\scriptstyle{n=2:}$}; \node(k3) at (-1,-3) {$\scriptstyle{n=3:}$}; \node(k4) at (-1,-4) {$\scriptstyle{n=4:}$}; \node(k5) at (-1,-5) {$\scriptstyle{n=5:}$}; \node(k6) at (-1,-6) {$\scriptstyle{n=6:}$}; \node(k7) at (-1,-7) {$\scriptstyle{n=7:}$}; \node(k8) at (-1,-8) {$\scriptstyle{n=8:}$}; \node(k9) at (-1,-9) {$\scriptstyle{n=9:}$}; \node(k9) at (-1,-10) {$\scriptstyle{n=10:}$}; \node[wV, label=above:{\small$0$},label=below:{\color{red}{\small{\bf1}}}] (V00) at (0,0) {}; \node[wV, label=above:{\small$1$},label=below:{\color{red}{\small{\bf1}}}] (V11) at (1,-1) {}; \node[wV, label=above:{\small$0$},label=below:{\color{red}{\small{\bf1}}}] (V20) at (0,-2) {}; \node[wV, label=above:{\small$2$},label=below:{\color{red}{\small{\bf1}}}] (V22) at (2,-2) {}; \node[wV, label=above:{\small$1$},label=below:{\color{red}{\small{\bf2}}}] (V31) at (1,-3) {}; \node[wV, label=above:{\small$3$},label=left:{\color{red}{\small{\bf1}}}] (V33) at (3,-3) {}; \node[wV, label=above:{\small$0$},label=below:{\color{red}{\small{\bf2}}}] (V40) at (0,-4) {}; \node[wV, label=above:{\small$2$},label=below:{\color{red}{\small{\bf3}}}] (V42) at (2,-4) {}; \node[wV, label=above right:{\small$4'$},label=left:{\color{red}{\small{\bf1}}}] (V44p) at (3,-4) {}; \node[wV, label=above:{\small$4$},label=below:{\color{red}{\small{\bf1}}}] (V44) at (4,-4) {}; \node[wV, label=above:{\small$1$},label=below:{\color{red}{\small{\bf5}}}] (V51) at (1,-5) {}; \node[wV, label=above:{\small$3$},label=left:{\color{red}{\small{\bf5}}}] (V53) at (3,-5) {}; \node[wV, label=right:{\small$5$},label=left:{\color{red}{\small{\bf1}}}] (V55) at (5,-5) {}; \node[wV, label=above:{\small$0$},label=below:{\color{red}{\small{\bf5}}}] (V60) at (0,-6) {}; \node[wV, label=above:{\small$2$},label=below:{\color{red}{\small{\bf10}}}] (V62) at (2,-6) {}; \node[wV, label=above right:{\small$4'$},label=left:{\color{red}{\small{\bf5}}}] (V64p) at (3,-6) {}; \node[wV, label=above:{\small$4$},label=below:{\color{red}{\small{\bf6}}}] (V64) at (4,-6) {}; \node[wV, label=above:{\small$6$},label=below:{\color{red}{\small{\bf1}}}] (V66) at (6,-6) {}; \node[wV, label=above:{\small$1$},label=below:{\color{red}{\small{\bf15}}}] (V71) at (1,-7) {}; \node[wV, label=right:{\small$3$},label=left:{\color{red}{\small{\bf21}}}] (V73) at (3,-7) {}; \node[wV, label=right:{\small$5$},label=left:{\color{red}{\small{\bf7}}}] (V75) at (5,-7) {}; \node[wV, label=above:{\small$0$},label=below:{\color{red}{\small{\bf15}}}] (V80) at (0,-8) {}; \node[wV, label=above:{\small$2$},label=below:{\color{red}{\small{\bf36}}}] (V82) at (2,-8) {}; \node[wV, label=above right:{\small$4'$},label=left:{\color{red}{\small{\bf21}}}] (V84p) at (3,-8) {}; \node[wV, label=above:{\small$4$},label=below:{\color{red}{\small{\bf28}}}] (V84) at (4,-8) {}; \node[wV, label=above:{\small$6$},label=below:{\color{red}{\small{\bf7}}}] (V86) at (6,-8) {}; \node[wV, label=above:{\small$1$},label=below:{\color{red}{\small{\bf51}}}] (V91) at (1,-9) {}; \node[wV, label=right:{\small$3$},label=left:{\color{red}{\small{\bf85}}}] (V93) at (3,-9) {}; \node[wV, label=right:{\small$5$},label=below:{\color{red}{\small{\bf35}}}] (V95) at (5,-9) {}; \node[wV, label=above:{\small$0$},label=below:{\color{red}{\small{\bf51}}}] (V100) at (0,-10) {}; \node[wV, label=above:{\small$2$},label=below:{\color{red}{\small{\bf136}}}] (V102) at (2,-10) {}; \node[wV, label=above right:{\small$4'$},label=below:{\color{red}{\small{\bf85}}}] (V104p) at (3,-10) {}; \node[wV, label=above:{\small$4$},label=below:{\color{red}{\small{\bf120}}}] (V104) at (4,-10) {}; \node[wV, label=above:{\small$6$},label=below:{\color{red}{\small{\bf35}}}] (V106) at (6,-10) {}; \draw[line width=4pt,blue!20] (V00) to (V11) to (V22) to (V33) to (V44) to (V55) to (V66); \draw[line width=4pt,blue!20] (V33) to (V44p); \draw[line width=0.7] (V00) to (V11) to (V20) to (V31) to (V40) to (V51) to (V60) to (V71) to (V80) to (V91) to (V100); \draw[line width=0.7] (V11) to (V22) to (V31) to (V42) to (V51) to (V62) to (V71) to (V82) to (V91) to (V102); \draw[line width=0.7] (V33) to (V42) to (V53) to (V62) to (V73) to (V82) to (V93) to (V102); \draw[line width=0.7] (V22) to (V33) to (V44) to (V55) to (V66); \draw[line width=0.7] (V44) to (V53) to (V64) to (V73) to (V84)to (V93) to (V104); \draw[line width=0.7] (V55) to (V64) to (V75) to (V84) to (V95) to (V104); \draw[line width=0.7] (V66) to (V75) to (V86) to (V95) to (V106); \draw[line width=0.7] (V33) to (V44p) to (V53) to (V64p) to (V73) to (V84p) to (V93) to (V104p); \end{tikzpicture}\end{matrix} $$ \end{SVG}

In another exciting collaboration, Georgia worked with Persi Diaconis, Martin Liebeck, and Pham Huu Tiep (see BDLT20) at MSRI to use fusion matrices to analyze families of Markov chains. They studied walks in a similar manner to the case pictured in Figure 8 above, except now using groups and quantum groups like

instead of the octahedral group used in Figure 8.

The game is similar to walking on graphs with representations and the McKay correspondence. You start with an empty mixing bowl, choose a small representation, put it in the bowl, and hand it to the next cook. The second cook chooses a small representation to tensor with, and mixes it into the bowl (i.e., calculates the tensor product with what is already there) and hands it on to the next cook in line. This process continues …, and there’s one person at the restaurant (Persi Diaconis) who always wants to know when the food is going to arrive, i.e., how long it takes for all this mixing and cooking to get to the stationary state.

There are several finicky issues that have to be dealt with:

(a)

In characteristic