Modified Hall–Littlewood polynomials and characters of affine Lie algebras Nicholas Alexander Booth Bartlett BSc. Hons. I A thesis submitted for the degree of Doctor of Philosophy at The University of Queensland in 2013 School of Mathematics and Physics Abstract In the late 60’s and early 70’s V. Kac and R. Moody developed a theory of generalised Lie algebras which now bears their name. As part of this theory, Kac gave a beautiful generalisationofthefamousWeylcharacterformulaforthecharactersofintegrablehighest weightmodules,raisingtheclassicalresulttothelevelofKac–Moodyalgebras. TheWeyl– Kac character formula, as it is now known, is a powerful statement that preserves all of the desireable properties of Weyl’s formula. However, there is one drawback that also remains. Kac’s result formulates the characters of Kac–Moody algebras as an alternating sum over the Weyl group of the underlying affine root system. This inclusion-exclusion type representation obscures the natural positivity of these characters. The purpose of this thesis is to provide manifestly positive (that is, combinatorial) representations for the characters of affine Kac–Moody algebras. In our pursuit of this task, we have been partially successful. For 1-parameter families of weights, we derive combinatorialformulasofso-calledLittlewoodtypeforthecharactersofaffineKac–Moody (2) (1) (2) algebrasoftypesA andC . FurthermoreweobtainasimilarresultforD , although 2n n n+1 this relies on an as-yet-unproven case of the key combinatorial q-hypergeometric identity underlying all of our character formulas. Our approach employs the machinery of basic hypergeometric series to construct q- series identities on root systems. Upon specialisation, one side of these identities yields the above-mentioned characters of affine Kac–Moody algebras in their representation pro- vided by the Weyl–Kac formula. The other side, however, leads to combinatorial sums of Littlewood type involving the modified Hall–Littlewood polynomials. These polynomials form an important family of Schur-positive symmetric functions. This thesis is divided into two parts. The first part contains three chapters, each delivering a brief survey of essential classical material. The first of these chapters treats the theory of symmetric functions, with special emphasis on the modified Hall–Littlewood polynomials. The second chapter provides a short introduction to root sytems and the Weyl–Kac formula. The introductory sequence concludes with a chapter on basic hyper- geometric series, highlighting the Bailey lemma. All of our original work towards Littlewood-type character formulas is contained in Part II. This work is broken down into four chapters. In the first chapter, we use Milne and Lilly’s Bailey lemma for the C root system n to derive a C analogue of Andrews’ celebrated q-series transformation. It is from this n transformation that we will ultimately extract our character formulas. Inthesecondchapterwedevelopasubstantialamountofnewmaterialforthemodified Hall–Littlewood polynomials Q(cid:48) . In order to transform one side of our C Andrews λ n transformation into Littlewood-type combinatorial sums, we need to prove a novel q- hypergeometric series identity involving these polynomials. We (partially) achieve this by i first proving a new closed-form formula for the Q(cid:48) . For this proof in turn we rely heavily λ on earlier work by Jing and Garsia. The highlight of our work is the third chapter, where we bring together all of our prior results to prove our new combinatorial character formulas. The most interesting part of the calculations carried out in this section is a bilateralisation procedure which transforms unilateral basic hypergeometric series on C into bilateral series which exhibit the full n affine Weyl group symmetry of the Weyl–Kac character formula. The fourth and final chapter explores specialisations of our character formulas, result- ing in many generalisations of Macdonald’s classical eta-function identities. Some of our formulasalsogeneralisefamousidentitiesfrompartitiontheoryduetoAndrews,Bressoud, Go¨llnitz and Gordon. ii Declaration by author This thesis is composed of my original work, and contains no ma- terial previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis. I have clearly stated the contribution of others to my thesis as a whole, including statistical assistance, survey design, data analy- sis, significant technical procedures, professional editorial advice, andanyotheroriginalresearchworkusedorreportedinmythesis. The content of my thesis is the result of work I have carried out sincethecommencementofmyresearchhigherdegreecandidature and does not include a substantial part of work that has been sub- mitted to qualify for the award of any other degree or diploma in any university or other tertiary institution. I have clearly stated which parts of my thesis, if any, have been submitted to qualify for another award. I acknowledge that an electronic copy of my thesis must be lodged with the University Library and, subject to the General Award Rules of The University of Queensland, immediately made avail- able for research and study in accordance with the Copyright Act 1968. Iacknowledgethatcopyrightofallmaterialcontainedinmythesis resides with the copyright holder(s) of that material. Where ap- propriate I have obtained copyright permission from the copyright holder to reproduce material in this thesis. iii Publications during candidature [1] N. Bartlett and S. O. Warnaar, Hall–Littlewood polynomials and characters of affine Lie algebras, arXiv:1304.1602. Publications included in this thesis Almostallofthefindingsof[1]areincludedthisthesis. Thebreakdownofworkundertaken in the production of [1] is as follows. Contributor Statement of Contribution N. Bartlett Conception and design 50 % Analysis and interpretation 50 % Drafting and critical review 50 % S. O. Warnaar Conception and design 50 % Analysis and interpretation 50 % Drafting and critical review 50 % Contributions by others to the thesis No further contributions by others. Statement of parts of the thesis submitted to qualify for the award of another degree None. iv Acknowledgments To my parents, Graeme L. and Elizabeth S. Bartlett You are a reliable well-spring of courage and wisdom. I would call you up, deeply troubled, and you would wave your hands, sprinkle a little sage advice and in the span of a few minutes I would be feeling hale and hearty again. I now know that these are the rites of an ancient form of parental-witchcraft, in which the anxiety is drawn from the child and taken into oneself. I’m sorry you were called upon to suffer so often, even in the midst of your own substantial individual trials. I’m sorry further that I won’t yet relieve you of your torment. You must be worried about my long-anticipated bicycle-touring trip, especially with our rather unfortunate family record for hospitalisation during overseas travel. I’d like you to bear in mind that you all survived, each with a complete recovery. Expect that I’ll do the same (most likely, better). Now that this thesis comes to an end, recall your slight frustration with my lack of focus in highschool. I take this opportunity to point out that had I scored slightly better grades, I would have made the hideous error of entering a degree in Journalism. Thanks again for your steady support. You are my loudest barrackers, my most trusted confidants, my battered shields to anxiety, and most importantly, my pri- mary source of delicious morale-boosting goods delivered by post. To my supervisor, Professor S. Ole Warnaar I cannot hope to ever repay you for your kind instruction. Needless to say, I am ecstatic that this thesis will soon be behind me, but I have greatly enjoyed your tutelage and so it is with some sadness that I finish. There is also some trepidation, since I still have no clear idea of what I will do next. One observes your successes and wonders if they might be reproduced. In my case, I think not. In spite of your fine example and sharp mentoring, I will not be a professor of mathematics. Even with your veteran coaching and wise advice during my marathon training (perhaps too often brazenly ignored), a career as a professional athlete is not on the table either. It seems then that the determination of a worthy goal demands of me a little morecreativity. Fortunately,underyourdirectionI’velearnedthatthisisessentially a matter of persistent effort, and that’s easy! Althoughourpathssoondiverge, knowthatyourinfluencewillnotleaveme. Iknow that whatever I do, I must find it in me to do it with all the boldness, exactitude and sheer joy that you do mathematics. v Keywords Hall–Littlewood polynomials, symmetric functions, character identities ANZSRC Classification 010101 Algebra and Number Theory 50% 010104 Combinatorics and Discrete Mathematics 50% FoR Classification 0101 Pure mathematics 100% vi Contents Introduction ix Preliminary material 1 1 Symmetric Functions 1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Partitions and compositions . . . . . . . . . . . . . . . . . . . . . . . 1 Young tableaux and Kostka numbers . . . . . . . . . . . . . . . . . . 4 Classical symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . 5 Standard bases of the ring of symmetric functions . . . . . . . . . . . 6 The Schur functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 The Hall inner product . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Hall–Littlewood polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The modified Hall–Littlewood polynomials Q(cid:48) . . . . . . . . . . . . . 15 λ The notation of λ-rings and the polynomials Q . . . . . . . . . . . . 18 λ 2 Characters of affine Kac–Moody Lie algebras 21 Finite root systems and the Weyl character formula . . . . . . . . . . . . . 22 Finite root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 The Weyl character formula . . . . . . . . . . . . . . . . . . . . . . . 27 Littlewood-type sums . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Characters of affine Kac–Moody Lie algebras . . . . . . . . . . . . . . . . . 31 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 The Weyl–Kac Character formula . . . . . . . . . . . . . . . . . . . . 36 3 Basic hypergeometric series 42 Basic hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Bailey’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Identities of Rogers–Ramanujan type . . . . . . . . . . . . . . . . . . 48 vii Combinatorial character formulas 53 4 The C Andrews transformation 53 n C basic hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . . 53 n The C Bailey lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 n The C Andrews Transformation . . . . . . . . . . . . . . . . . . . . . . . 56 n 5 The modified Hall–Littlewood polynomials 60 An explicit formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 The Rogers–Szeg˝o polynomials . . . . . . . . . . . . . . . . . . . . . . . . 66 A conjectural q-hypergeometric identity for a Littlewood-type sum . . . . . 68 6 Combinatorial character formulas 74 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Proof of Theorem 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 The right-hand side . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 The left-hand side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 7 Generalised Macdonald eta-function identities 88 Generalised eta-function identities . . . . . . . . . . . . . . . . . . . . . . . 89 Type B(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 n Type C(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 n Type A(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2n Type A(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2n−1 Type D(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 n+1 Details of specialisation procedure . . . . . . . . . . . . . . . . . . . . . . . 96 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Example: specialisation for type C(1) . . . . . . . . . . . . . . . . . . 98 n Bibliography 99 viii Introduction In the late 60’s and early 70’s V. Kac and R. Moody developed a theory of gener- alised Lie algebras which now bears their name. As part of this theory, Kac gave a beautiful generalisation of the famous Weyl character formula for the characters of integrable highest weight modules, raising the classical result to the level of Kac– Moody algebras. The Weyl–Kac character formula, as it is now known, is a powerful statement that preserves all of the desireable properties of Weyl’s formula. However, there is one drawback that also remains. Kac’s result formulates the characters of Kac–Moody algebras as an alternating sum over the Weyl group of the underlying affine root system. This inclusion-exclusion type representation obscures the natural positivity of these characters. The purpose of this thesis is to provide manifestly positive (that is, combinato- rial) representations for the characters of affine Kac–Moody algebras. In our pursuit of this task, we have been partially successful. For 1-parameter families of weights, we derive combinatorial formulas of so-called Littlewood type for the characters of affine Kac–Moody algebras of types A(2) and C(1). Furthermore we obtain a similar 2n n result for D(2) , although this relies on an as-yet-unproven case of the key combina- n+1 torial q-hypergeometric identity underlying all of our character formulas. Our approach employs the machinery of basic hypergeometric series to construct q-series identities on root systems. Upon specialisation, one side of these identities yields the above-mentioned characters of affine Kac–Moody algebras in their rep- resentation provided by the Weyl–Kac formula. The other side, however, leads to combinatorial sums of Littlewood type involving the modified Hall–Littlewood poly- nomials. These polynomials form an important family of Schur-positive symmetric functions. This thesis is divided into two parts. The first part contains three chapters, each delivering a brief survey of essential classical material. The first of these chapters treats the theory of symmetric functions, with special emphasis on the modified Hall–Littlewood polynomials. The second chapter provides a short introduction to root sytems and the Weyl–Kac formula. The introductory sequence concludes with a chapter on basic hypergeometric series, highlighting the Bailey lemma. ix