TWO PROBLEMS IN REPRESENTATION THEORY: AFFINE LIE ALGEBRAS AND ALGEBRAIC COMBINATORICS By ALEJANDRO GINORY A dissertation submitted to the School of Graduate Studies Rutgers, The State University of New Jersey In partial fulfillment of the requirements For the degree of Doctor of Philosophy Graduate Program in Mathematics Written under the direction of Siddhartha Sahi And approved by New Brunswick, New Jersey May, 2019 ABSTRACT OF THE DISSERTATION Two Problems in Representation Theory: Affine Lie Algebras and Algebraic Combinatorics By ALEJANDRO GINORY Dissertation Director: Siddhartha Sahi In this dissertation, we investigate two topics with roots in representation theory. The first topic is about twisted affine Kac-Moody algebras and vector spaces spanned by their characters. Specifically, the space spanned by the characters of twisted affine Lie algebras admit the action of certain congruence subgroups of SL(2; Z). By embedding the characters in the space spanned by theta functions, we study an SL(2; Z)-closure of the space of characters. Analogous to the untwisted affine Lie algebra case, we construct a commutative associative algebra (fusion algebra) structure on this space through the use of the Verlinde formula and study important quotients. Unlike the untwisted cases, some of these algebras and their quotients, which relate to the trace of diagram automorphisms on conformal blocks, have negative structure constants with respect to the (usual) basis indexed by the dominant integral weights of the Lie algebra. We give positivity conjectures for the new structure constants and prove them in some illuminating cases. We then compute formulas for the action of congruence subgroups on these character spaces and give explicit descriptions of the quotients using the affine Weyl group. The second topic concerns algebraic combinatorics and symmetric functions. In statistics, zonal polynomials and Schur functions appear when taking integrals over ii certain compact Lie groups with respect to their associated Haar measures. Recently, a conjecture, related to certain integrals of statistical interest, was proposed by D. Richards and S. Sahi. This conjecture asserts that certain linear combinations of Jack polynomials, a one-parameter family of symmetric polynomials that generalizes the zonal and Schur polynomials, are non-negative when evaluated over a certain cone. In the second part of this dissertation, we investigate these conjectures for Schur polyno- mials and give a refined version of the conjecture. In addition, we prove some cases and arrive at certain seemingly new combinatorial results. In an important instance, we give an analogous result for Jack polynomials. iii Acknowledgements First, I would like to express my deepest gratitude to my advisor, Siddhartha Sahi, for introducing me to and guiding me through so many areas in mathematics. I am especially grateful for his intellectual generosity, abundant patience, and razor sharp wit. It has been a great honor to work with him and witness his tireless curiosity up close. I would like to thank the dissertation committee members Jim Lepowsky, Yi-Zhi Huang, and Jiuzu Hong for their help and comments on my thesis. I would further like to thank Jim for the many hours of discussion of many interesting problems and his guidance in my graduate studies in general. Thank you to Yi-Zhi for many useful conversations that opened up new avenues of inquiry. Thank you to Jiuzu for great discussions and hosting me during my visit to the University of North Carolina at Chapel Hill. I would also like to thank all of those professors who took time to discuss mathe- matics with me. Let me mention a few. Thanks to Chris Woodward, who was generous enough to start working with me as a first year graduate student. Thanks to Vladimir Retakh, who was willing to listen to any idea I would come up with throughout the years. Thanks to Lisa Carbone, for also serving on my oral qualifying examination committee and discussing potential projects we might work on. Thanks to Doron Zeil- berger, whose opinions gave me a different perspective on many topics and who helped me appreciate the relevance of experimental mathematics. Thanks to Roger Nussbaum who honored me by taking interest in a certain discovery that I came across and was patient enough to guide me through the wonderful, yet alien, world of non-linear anal- ysis. Thanks to Angela Gibney, who mentored me, challenged me, and worked with me intensely in the short time we have known each other. Let me also thank some of the iv important professors who helped me during my undergraduate years: Tedi Draghici, Laura De Carli, Miroslav Yotov, Gueo Grantcharov, and Thomas Leness. A special thanks goes out to all of my peers and colleagues for helping me grow while staying sane. In particular, thanks to Ed Karasiewicz, Zhuohui Zhang, Sjuvon Chung, Pat Devlin, Fei Qi, Richard Voepel, Rebecca Coulson, Jorge Cantillo, Mingjia Yang, Matthew Russell, Doug Schultz, Semeon Artamonov, Johannes Flake, Robert Laugwitz, Jason Saied, Sven Moeller, Jongwon Kim, Henry Zorrilla, and Eric Wawerczyk. I could go on, but I won't. I would like to thank Kafung Mok for her endless support and patience during these years. I could not do anything without her. She was and continues to be the light of my life. I would also like to thank my family for their support and encouragement throughout my journey. Finally, I would like to thank Alejandro Ortiz-Burgos and his wife Dagmar, kind strangers who helped a lowly waiter discover mathematics. As for non-human `actors', I would like to thank the warmth of my hometown Miami for recharging me every time I visited, the summer season in New Jersey for being the only reprieve I got from the dreariness of the rest of the year, and the Middlesex Greenway, a former railway track converted into a beautifully green trail, where I would attempt to remain healthy. v Table of Contents Abstract :::::::::::::::::::::::::::::::::::::::: ii Acknowledgements ::::::::::::::::::::::::::::::::: iv 1. Introduction ::::::::::::::::::::::::::::::::::: 1 2. Fusion Algebras, the Verlinde Formula, and Twisted Affine Kac-Moody Algebras :::::::::::::::::::::::::::::::::::::::: 4 2.1. Introduction . .4 2.1.1. Structure of Chapter . 10 2.2. Preliminaries . 11 2.2.1. Affine Kac-Moody Lie Algebras . 11 2.2.2. Integrable Highest Weight Modules . 14 2.2.3. Theta Functions, Modular Invariance, and Characters . 15 2.3. Verlinde Algebras of Untwisted Affine Algebras . 19 2.4. Verlinde Algebras of Twisted Affine Algebras . 24 (2) 2.5. Verlinde Algebras for A2` .......................... 29 2.6. Applications . 33 2.6.1. Quotients of Vk(g).......................... 33 2.6.1.1. More Negative Structure Constants . 37 (2) 2.6.2. The A2` Quotient Structure . 38 2.6.3. Congruence Group Action . 42 2.7. Some Data . 44 3. Richards-Sahi Conjectures for Schur Polynomials :::::::::::: 52 3.1. Introduction . 52 vi 3.2. Preliminaries . 54 3.2.1. Partitions . 54 3.2.2. Symmetric Polynomials . 56 r 3.3. αµ Conjecture . 58 3.4. Special Cases of Richards-Sahi Conjecture . 59 3.4.1. Case µ = ni .............................. 59 3.4.2. Case µ = m11n ............................ 61 3.5. Another Partial Order on Partitions . 65 3.6. Certain Chains of Partitions . 70 3.7. Refinement of Conjecture . 73 3.7.1. Positive Functions γµλ ........................ 74 3.8. Kostka-Content Identities . 77 3.8.1. An Inductive Argument . 78 3.8.2. Case λ = µ .............................. 80 3.8.3. Jack Hook-Length Recursion Formula . 81 References ::::::::::::::::::::::::::::::::::::::: 88 vii 1 Chapter 1 Introduction Representation theory is a broad field with many beautiful branches and applications. In this dissertation, we will explore two topics in representation theory that at first glance seem very far apart, but at their core are concerned with representations of Lie algebras. Both address conjectures involving mysterious identities, giving concrete answers in certain important families. Both give rise to even deeper questions about their connections to other fields. Finally, both involve computational and experimental techniques, though ultimately the results are purely conceptual. This dissertation has two main stand-alone parts, Chapters 2 and 3, each an essay in its own right. In first part, we try to determine what sort of hidden algebraic structures exist among certain classes of representations of twisted affine Kac-Moody Lie algebras. More precisely, we try to find an analogue to the Verlinde fusion algebras, which exist for untwisted affine Kac-Moody algebras, for the twisted case. In the second part, we investigate curious conjectures about positivity arising from statistics, but with representation theoretic connections, give partial results confirming them in many cases, and provide a further refinement of the conjectures. The identities are formulated in the generality of Jack polynomials, and, in this work, we focus on the special case of Schur polynomials, where the conjectures are no easier to prove. Many of the results for Schur polynomials generalize easily and directly to Jack polynomials, and the Jack polynomial case will be the object of future study. Each part has an introduction which will motivate and describe the results in that chapter, but let us a give a broad outline of what is contained there. In the first part, we give a background on Verlinde fusion algebras for untwisted affine Kac-Moody algebras and then discuss what we do for the twisted Kac-Moody algebras. Specifically, 2 we study an embedding of the space spanned by the characters of twisted Kac-Moody algebras into a SL2(Z)-module and then immediately show that, in most cases, this space is isomorphic to a Verlinde fusion algebra. One case stands out among the rest, (2) that is the case of Kac-Moody algebras of type A2` ; where we show that this gives rise to a new fusion algebra. This fusion algebra has special properties that differ from the usual Verlinde fusion algebras.