c [2016] Bud B. Coulson ALL RIGHTS RESERVED

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1 c 206 Bud B. Coulson ALL RIGHTS RESERVED

2 AN AFFINE WEYL GROUP INTERPRETATION OF THE MOTIVATED PROOFS OF THE ROGERS-RAMANUJAN AND GORDON-ANDREWS-BRESSOUD IDENTITIES BY BUD B. COULSON A dissertation submitted to te Graduate Scool New Brunswick Rutgers, Te State University of New Jersey in partial fulfillment of te requirements for te degree of Doctor of Pilosopy Graduate Program in Matematics Written under te direction of James Lepowsky and approved by New Brunswick, New Jersey May, 206

3 ABSTRACT OF THE DISSERTATION An affine Weyl group interpretation of te motivated proofs of te Rogers-Ramanujan and Gordon-Andrews-Bressoud identities by Bud B. Coulson Dissertation Director: James Lepowsky A motivated proof of te Rogers-Ramanujan identities was given by G. E. Andrews and R. J. Baxter. Tis proof was generalized to te odd-moduli case of Gordon s identities by J. Lepowsky and M. Zu, and later to te even-moduli case of te Andrew-Bressoud identities by S. Kanade, Lepowsky, M. C. Russell and A. Sills. We present a reinterpretation of tese proofs, wit new motivation coming from te affine Weyl group of sl(2). ii

4 Acknowledgements I would like to express my sincere gratitude to my advisor Jim Lepowsky for is continual support of my researc and for is patience, motivation, and immense knowledge. His guidance as been invaluable, and tis tesis would not exist witout is aid and encouragement. I would also like to tank te rest of my tesis committee: Yi-Zi Huang, Lisa Carbone, and Corina Calinescu, for teir insigtful comments and advice. Finally, I would like to tank my family: my parents and siblings for teir love and support; and my wonderful, loving, selfless wife Becky for being wit me always. iii

5 Dedication Dedicated to my son, Iggy. iv

6 Table of Contents Abstract ii Acknowledgements iii Dedication iv. Introduction Background material Affine Lie algebras Partitions Rogers-Ramanujan case - new interpretation Introduction Definitions and background Closed formulas Empirical Hypotesis Combinatorics Gordon-Andrews case - new interpretation Introduction Definitions and background Closed formulas Empirical Hypotesis Combinatorics Andrews-Bressoud case - new interpretation Introduction v

7 5.2. Definitions and background Closed formulas Empirical Hypotesis Combinatorics References vi

8 Capter Introduction Te classical Rogers-Ramanujan partition identities state tat were m, m 0, ±2 (mod 5) m, m 0, ± (mod 5) q m = n 0 q m = n 0 d (n)q n, (.) d 2 (n)q n, (.2) d (n) = te number of partitions of n for wic adjacent parts ave difference at least 2 and d 2 (n) = te number of partitions of n for wic adjacent parts ave difference at least 2 and no part is equal to Te product sides (i.e., te left-and sides above) of te Rogers-Ramanujan identities enumerate te partitions wose parts obey certain restrictions modulo 5, and te sum sides (te rigt-and sides) enumerate te partitions wit certain difference-two and initial conditions. Generalizations of te Rogers-Ramanujan identities for all odd moduli were discovered by B. Gordon G and G. E. Andrews A. Analogous identities for te even moduli of te form 4k + 2 were discovered by Andrews in A2 and A3, and subsequently, for all te even moduli, by D. M. Bressoud in Br. In AB, Andrews and A. Baxter gave an interesting motivated proof of tese two Rogers- Ramanujan identities, a variant of one of te original proofs by Rogers and Ramanujan and of an earlier proof by Baxter imself. In teir proof, tey explained te difference-two condition

9 2 appearing in te sum sides directly from te product sides, and in doing tis, tey were able to bot motivate te expressions on te sum sides and prove te two identities. Teir metod was to start by rewriting te product sides of te two identities using te classical Jacobi triple product identity (as in done is most proofs of te Rogers-Ramanujan identities), ten to take certain combinations of tese series to generate an infinite tower of series. Once notices empirically tat tese iger series converge to in a suitable sense. Tey called tis assertion te Empirical Hypotesis, and tey were able to prove it by giving closed-formed expressions for te iger series. Ten, solving for te base series in terms of tese iger series exactly yields te partition conditions present on te sum sides. Recently, J. Lepowsky togeter wit students and collaborators ave given a series of analogous motivated proofs for various identities of Rogers-Ramanujan type, in wic te motivation as been similar in spirit to te motivation in AB. Correspondingly, we will use te term motivated proof as a tecnical term. In particular, in LZ, Lepowsky and M. Zu gave a motivated proof of te Gordon-Andrews generalization of te Rogers-Ramanujan identities, and in KLRS, S. Kanade, Lepowsky, M. C. Russell and A. Sills gave a motivated proof of te Andrews-Bressoud identities. It is well known tat partition identities of Rogers-Ramanujan type are closely related to te representation teory of vertex operator algebras. Early vertex operator teory was in fact motivated by te successful attempt to realize te combinatorial sum sides of identities of tis type - exibiting tem as te graded dimensions of vector spaces constructed from certain natural vertex operators. In a series of papers (LW2-LW4), Lepowsky and R.L. Wilson accomplised tis using te teory of principally twisted Z-operators, built out of te twisted vertex operators starting from LW. In fact, te Z-algebraic structure developed in LW2-LW4 gave a vertexoperator-teoretic interpretation of te wole family of Gordon-Andrews-Bressoud identities, as well as a vertex-operator-teoretic proof of te Rogers-Ramanujan identities, in te context of te affine Lie algebra A (). For te cases beyond Rogers-Ramanujan, A. Meurman and M. Primc, in MP, extended tis vertex-operator-teoretic interpretation to a full proof of te iger identities. Tis Z-algebra viewpoint (or an equivalent formulation) was later used by K. Misra Mi-Mi4, M. Mandia Ma, C. Xie X, S. Capparelli Cap-Cap2, M. Tamba - Xie TX, M. Bos - K. Misra BM, and D. Nandi N to give furter interpretations and proofs of tese same identities, as well as to study new identities, in te context of a wider range of affine Lie algebras. A very different

10 3 vertex-algebraic approac to te sum sides of te Rogers-Ramanujan and Gordon-Andrews identities was developed in CLM, CLM2, CalLM and CalLM based on untwisted intertwining operators. Tese connections ave been major incentives in seeking out motivated proofs. However, in all of te motivated proof papers referenced above, te proofs ave been based entirely on te manipulation of q-series, wile all of te vertex-operator-algebraic proofs ave made use of deep vertex algebraic teory. Tis distinction between te two approaces as made it difficult to understand motivated proofs from a strictly vertex-operator teoretic viewpoint. Te present work was inspired by our desire to approac tis ultimate goal, and we ave succeeded in making a significant step in tis direction. We coose to view te Jacobi triple product identity as te denominator identity for te affine Lie algebra A () = sl(2), in order to recast te q-series entering into te proofs of te Gordon-Andrews-Bressoud identities in terms of te affine Weyl group of sl(2). It turns out tat tis leads to surprising insigts into te nature of tese series and into te algebraic and geometric structure underlying te motivated proofs of tese identities. In particular, most parts of te motivated proofs are made sorter and take on a muc more natural form, and te nature of certain contributions to te series becomes muc more transparent. Wile te proofs are still based on q-series manipulation, te ad oc manipulations used previously become motivated transformations tat come directly from te Coxeter group. Tis work sould elp to bridge te gap between te motivated proof papers and te vertex-algebraic papers mentioned above. We expect tat tis approac can be generalized to iger rank affine Lie algebras. Applying tis analysis to iger-rank affine Lie algebras sould lead to many new identities of Rogers-Ramanujan type, as well as to new proofs of known identities. In tis work, first we treat te special test case of te Rogers-Ramanujan identities. Once aving gone troug all te proofs ere, we andle te more general case of te Gordon-Andrews identities. Finally, we deal wit te Andrews-Bressoud identities, wic exibit some new beavior but follow te same general paradigm.

11 4 Capter 2 Background material 2. Affine Lie algebras We start wit recalling basic teory of affine Lie algebras, and specifically information related to te smallest affine Lie algebra A () = sl(2). Tis material is covered in many introductory texts - see for example Ca or K. After reviewing some basic material, we will specify wat we will be using in tis work. Te general Weyl-Kac denominator formula states tat given an affine Lie algebra g, we ave ( ) l(w) e wρ ρ = ( e α ) (2.) α + were W is te associated affine Weyl group, + is te set of positive roots, and ρ is te fundamental weigt. For w W, te quantity wρ ρ will be extremely important to us. As is well known, tis expression is equal to te sum of te positive roots made negative by w (see for example Ca, Capter 20). Te standard notation for tis set of roots is w, and correspondingly, we will denote te wρ = ρ by te symbol w (were te absolute value is meant to indicate te sum of te enclosed set). Tis action can be computed explicitly from te action of te generators (basic reflections): for any root γ, s i γ = γ α i, γ i were α i, i are respectively te root and coroot corresponding to s i.

12 5 Remark 2.. Altoug we ave written te denominator formula above in its natural generality, in tis work, we will be exclusively concerned wit te concrete affine Lie algebra A (). Hereafter, wenever W appears, it is to be understood to refer to te Weyl group of tis particular affine Lie algebra (i.e., te affine Weyl group of A = sl(2)). Te group W can be viewed as te group of affine reflections on a one-dimensional lattice. It as generators s 0 and s (corresponding to te two simple roots, α 0 and α ), wit defining relations s 2 0 = s 2 = e (and no oters). Te sum δ = α 0 + α is te basic imaginary root, and it is well known tat tere are tree strings of positive roots: te imaginary roots kδ, k > 0 and te real roots α p + kδ, p = 0,, k 0 Wen working in tis context, given an element of te root lattice (for example, w ), we will denote its coordinates in te α 0, α basis by te subscripts 0,. In particular, we ave w = w 0 α 0 + w α (2.2) Remark 2.2. In tis work, we will deal extensively wit formal power series in a formal variable q (i.e., formal series wit non-negative integral powers of q), were te summands are indexed by elements of W. Te summand corresponding to w W will always be a polynomial in q, and te data making up tis polynomial will always be expressed in terms of te w quantities defined above. Tus, it is of te utmost importance to know te exact dependence of w on w, and ow it canges under certain transformations of W. We now compute te action of te generators s 0, s on w. First, we recall te basic actions

13 6 of of te Weyl group on te roots, wic follow directly from te formula given above: s 0 α 0 = α 0 (2.3) s 0 α = 2α 0 + α (2.4) s α 0 = α 0 + 2α (2.5) s α = α (2.6) Applying tese equations, we get: s0 w 0 = w w + (2.7) s0 w = w (2.8) s w 0 = w 0 (2.9) s w = 2 w 0 w +. (2.0) It is also direct from te definition tat e = 0 = 0α 0 + 0α Using tis initial condition and te above recursions, we see tat te values assumed by te w components are te triangle numbers. Specifically, te points w, w W trace out te parabola ( w 0 w ) 2 = w 0 + w (2.) in te α 0 α plane. To clarify te above, and for later use, at tis point we introduce special notation for Weyl group elements: for 0, let w 0 denote te Weyl element of lengt wose sortest expression (in te generators s 0, s ) starts wit s 0 and w denote te Weyl element of lengt wose sortest expression starts wit s. Notice tat w0 0 = w0 bot correspond to te identity element of te group. Explicitly, for 0, we write w 0 = w = lengt {}}{ s 0 s s 0 (2.2) lengt {}}{ s s 0 s (2.3)

14 7 Ten we can restate te conclusions above as w 0 = 0 w 0 = ( ) + 2 = w ( ) = 2 w 0 (2.4) (2.5) It follows tat w 0 = 0 w 0 + (2.6) w = w + 0 (2.7) Now we ave developed enoug background to revisit te denominator equation in te context of A (). It becomes ( ) l(w) e w = (e α 0, e δ ) (e α, e δ ) (e δ, e δ ) were (a; q) = m 0( aq m ) (te q-pocammer symbol). Te tecnical proofs in tis work all deal wit te manipulation of formal power series of te form described in Remark 2.2. Te series will be different for te different classes of identities we consider, but tey will all ave certain features in common. Our basic tools for manipulating tese series are te application of certain transformations of W, will wic act on te series as canges of index. Many times, we will need to make use of ow tese transformations affect te common features of te series. For convenience, at tis point we record tese results ere, and ten reference tem later as appropriate. Definition 2.3. Let s : W W be te map of left-composition wit te generator s. Tis map is an involution. In terms of te notation introduced above, tis map acts on elements of W by intercanging w 0 and w +. Its action on an expression of te form w is recorded above in (2.9), (2.0).

15 8 Definition 2.4. Let f : W W be te outer automorpism wic excanges te two generators s 0 and s. Tis map is an involution. In terms of te notation introduced above, tis map acts on elements of W by intercanging w 0 and w (in particular, it fixes te identity element e W ). Its action on an expression of te form w is to switc its components in formula (2.2). Notice tat s canges te lengt of eac element by ±, wile f( ) preserves te lengts of elements. Let w be an element of te affine Weyl group W. Consider te following expressions in te formal variable q: (q w q w r 0 ), (q w 0 q w r ) It follows from te discussion above tat te transformation s turns te first expression into (q 2 w 0 w + q w 0 r ) = q w 0 w + (q w 0 q w (r+) ) and te second into (q w 0 q 2 w 0 w + r ) = q w 0 w (q w q w 0 (r ) ) On te oter and, te transformation f will clearly just intercange tese two expressions. It follows tat: Corollary 2.5. Applying te transformations s, f( ) to W gives: s : (q w q w 0)(q w 0 q w )(q w q w 0 ) (q w 0 q w ) (q w q w 0)(q w 0 q w )(q w q w 0 ) (q w 0 q w )q (2 w 0 2 w +) f( ) : (q w q w 0)(q w 0 q w )(q w q w 0 ) (q w 0 q w ) (q w q w 0)(q w 0 q w ) (q w q w 0 ( ) )(q w q w 0 )

16 9 s : (q w q w 0)(q w 0 q w )(q w q w 0 ) (q w q w 0 ) (q w q w 0)(q w 0 q w ) (q w 0 q w )(q w 0 q w (+) ) q (2+)( w 0 w )+(+) f( ) : (q w q w 0)(q w 0 q w )(q w q w 0 ) (q w q w 0 ) (q w q w 0)(q w 0 q w )(q w q w 0 ) (q w q w 0 ) Remark 2.6. Te result above can be made to look even more natural: wen tese factors occur in our series, te values r in te exponents actually arise as te following expressions: w r 0 w r for tose factors in parenteses wit w 0 first, and w 0 r 0 + w 0 r for tose factors in parenteses wit w first. Since s acts on W by intercanging wr 0 wr+, and te map f intercanges w 0 r w r, we see tat te transformation of expressions calculated above work even on tis level of notation. Tis is anoter strong motivation for considering te series in tis ligt. Remark 2.7. Note in particular tat in two of te cases, te expression is only rescaled by a sign and/or power of q. Tis fact will prove to be vital in te proofs of te edge-matcing penomena. Remark 2.8. As as been indicated above, te series sums we consider will always be over te full Weyl group. Frequently, we visualize te elements of te Weyl group as living on te parabola traced out by te components of teir respective w values. Because of tis, it may seem natural to view tese sums as two-sided infinite, wit te identity element (corresponding to te vertex of te parabola) in te middle. However, wat turns out to be best is to view te sum as one-sided infinite, starting at w = e and continuing in a zig-zag up te parabola. Tis is akin to te te appearance of te series in LZ, but wit te following important and fundamental difference: in LZ, te sum was made infinite by folding te Weyl group in two and pairing. Essentially, eac summand in te series in LZ corresponds to te combination of two different Weyl elements. In our approac, we instead introduce a natural linear ordering on te Weyl group: starting wit w0 0 = e, we alternatingly apply te two transformations above in te order s, f( ). In terms of

17 0 our notation, te ordering of te Weyl elements is w 0 0, w, w 0, w 2, w 0 2, Remark 2.9. Of course, te analogous transformation s 0 could ave also been considered in addition to or instead of s. Te fact tat we prefer s over s 0 is an arbitrary coice, made to confirm to earlier works on motivated proofs. Te origin of tis distinction comes from te coice of specializations of formula (2.) we make to define initial selves of series below - te alternative coices (switcing te specialization in eac case) would correspond to working wit te s 0 transformation trougout. Remark 2.0. Altoug we ave used te language of te Lie algebra A () in discussing te material above, all of te analysis in tis work takes place solely on te level of te affine Weyl group. Te relevant underlying teory is tat of Coxeter groups, not tat of Lie algebras. 2.2 Partitions Wen we discuss te combinatorics of te sum sides of te identities, we will be using te following terminology concerning partitions: A partition of a non-negative integer n is a finite nonincreasing sequence of positive integers, written as π = (π,..., π t ) suc tat π + + π t = n. Eac π s is called a part of π. Te lengt l(π) of π is te number of parts in π, and given a positive integer p, te multiplicity m(p) of p in π is te number of parts of π equal to p. As is conventional, we say tat te integer 0 admits te unique partition into no parts (te empty partition). Given a sequence (a n ), n 0, te corresponding generating function is te formal power series a n q n n 0 in te formal variable q. In all of te identities we consider (and as was already seen in (.), (.2)), we will want to interpret te rigt-and sides as generating functions of partitions satisfying certain

18 restrictions. In oter words, te coefficient of q n will be te number of partitions of n obeying te given restrictions.

19 2 Capter 3 Rogers-Ramanujan case - new interpretation 3. Introduction As was mentioned in te introduction, te Rogers-Ramanujan identities were te first identities to be given a motivated proof (in AB). Here we give a brief summary of teir motivation and teir proof tecnique. Starting wit te product sides of te identities (.), (.2), one subtracts te second series (wic tey denoted G 2 (q)) from te first one (denoted G (q)), and divided by q to obtain a new formal series G 3 (q). Next, one forms G 4 (q) = (G 2 (q) G 3 (q))/q 2. One repeats tis process, giving G i (q) = (G i 2 (q) G i (q))/q i 2 for all i 3, and notices empirically tat for eac i, G i (q) is a power series in te formal variable q, it as constant term, and G i (q) is divisible by q i. Tis is te Empirical Hypotesis of Andrews-Baxter, and its trut easily leads to a proof te two Rogers-Ramanujan identities. To prove te Empirical Hypotesis directly from te product sides, one starts by transforming te two initial Rogers-Ramanujan product sides into alternating sums using te Jacobi triple product identity. By running tese alternating sums troug te recursion given above, one obtains alternating sum formulas for te next several G i series, and can conjecture and prove closed-formed formulas for all of te iger series. However, te calculations necessary to establis tese formulas (formal manipulation and reindexing of q-series) are essentially ad-oc. We will be following LZ and te appendices to CKLMQRS, wic refined tis argument. Tese papers incorporate concepts not used in AB, suc as placing series on selves, as we will review. Wile te basic logic of te proof below is te same as in tese earlier papers, we will go troug te motivated proof from scratc using our new perspective, igligting te differences between tese prior works and our new approac, and giving new motivation for eac step. Our

20 3 approac leans eavily on te observation first made by Lepowsky and S. Milne in LM, tat te Rogers-Ramanujan series arise as graded dimensions of te level-3 modules for te affine Lie algebra A (). 3.2 Definitions and background By specializing te quantities e α 0, e α in (2.) to appropriate powers of a single variable q, we obtain te product sides of te Rogers-Ramanujan identities (up to a factor), and can use te identity above to express tem in alternating-sum form. Specifically, specializing e α 0 q 3, e α q 2, we get (q 2, q 5 ) (q 3, q 5 ) (q 5, q 5 ) = (q, q) (q, q 5 ) (q 4, q 5 ) = ( ) l(w) q 3 w 0 +2 w 0 (3.) Specializing e α 0 q 4, e α q, we get (q, q 5 ) (q 4, q 5 ) (q 5, q 5 ) = (q, q) (q 2, q 5 ) (q 3, q 5 ) = ( ) l(w) q 4 w 0 + w 0 (3.2) Denote by F (q) te series (q, q) = n ( q n ) Dividing tese equations troug by F (q), we recognize on te left-and sides te Rogers-Ramanujan product expressions from (.), (.2). In te proofs, we will work wit te rigt-and sides, wic we denote as R (q) = F (q) R 2 (q) = F (q) ( ) l(w) q 3 w 0 +2 w ( ) l(w) q 4 w 0 + w respectively (we coose te symbol R instead of G for tese series to distinguis between tese and te more general Gordon-Andrews series of te next capter). We empasize tat tese two expressions are our definition of a zerot self of series, wic we will subsequently use to generate infinitely many iger selves of series.

21 4 Starting wit tese two series, we inductively define an infinite sequence of series by te equation R i+2 (q) = R i(q) R i+ (q) q i (3.3) As mentioned above, tis recursion was originally motivated (in AB) by te empirically evident fact tat te resulting R i series are always of te form + q i +, wit all positive coefficients. A new feature seen ere for te first time is tat tis recursion also aligns naturally to te recursion for te w terms. Tis provides a new motivation coming directly from te Coxeter group structure. We arrange tese series into selves of two series eac. Te two series on self j are R j+ and R j+2. In particular, te zerot self consists of te series R and R 2 given above, and te second series on eac self is te same as te first series on te next self. In order to use te recursion, it is necessary tat all te series on a given self must sare a similar form. We demonstrate ow tis works for te first few selves: Te zerot self consists of te two series R 0+ (q) = F (q) R 0+2 (q) = F (q) ( ) l(w) q 3 w 0 +2 w ( ) l(w) q 4 w 0 + w Substituting tese series into te recursion yields R +2 (q) = R 0+(q) R 0+2 (q) q = qf (q) ( ) l(w) (q w q w 0)q 3 w 0 + w To complete self j =, we need a similarly-saped formula for te R 2 series. Te natural ting to try is to ave te same factor in parenteses and ensure tat after distributing te parenteses, te positive part is identical wit our previous R 2 formula. Tis approac yields: R + (q) = F (q) ( ) l(w) (q w q w 0)q 4 w 0 In order for tis formula to be valid, we need te difference R 0+2 R + to be 0. Tis difference is te series F (q) ( ) l(w) q 5 w 0 wic is in fact 0, as can be seen by applying te cange of index w s w. Tis as te effect of canging te lengt by, but as no effect on te value of w 0. Hence te overall effect is to negate eac term of te series, wic in turn implies tat te wole sum is 0.

22 5 Next, we recursively compute R 2+2 (q) = R +(q) R +2 (q) q 2 = q 2 F (q) ( ) l(w) (q w q w 0)(q w 0 q w )q 3 w 0 To complete self j = 2, we need a similarly-saped formula for te R 3 series. Again, we attempt to matc te parenteses, wic gives R 2+ (q) = qf (q) ( ) l(w) (q w q w 0)(q w 0 q w )q 2 w 0 + w As before, in order to verify tat tis series expression is identical to te previous expression for R 3 (q), we need to consider te difference R +2 R 2+. Tis is given by q 2 F (q) ( ) l(w) (q w q w 0)q 2 w 0 +2 w Tis time, we apply te cange of index w f(w). Clearly te lengt of a Weyl group element is invariant under tis transformation, and its effect on w is to intercange te components w 0 and w. Hence te transformed expression is q 2 F (q) f(w) W ( ) l(w) (q w 0 q w )q 2 w +2 w 0 wic is once again precisely te negation of te original difference series. As above, tis implies tat te sum is 0, and tat te two forms of te R 3 series are bot valid. Notice tat in tis last computation, te bijection w f(w) fixes te identity element of te Weyl group, w = e. However, tis does not cause any problems: te summand corresponding to te identity element is 0 because te factor (q e 0 q e ) is (q 0 q 0 ) = 0. Altoug it is not readily apparent yet at tis early stage, it is also te case tat te powers of q appearing in te denominator before te sum can be nicely expressed in terms of Weyl data. Tis will be made explicit below in te general series formulas. 3.3 Closed formulas We ave te following remarkable closed-form formulas for tese series:

23 6 Teorem 3.. Let j 0 and i =, 2. If j = 2 is even, ten R 2+i (q) = 2 F (q) q w0 (i ) 0 If j = 2 + is odd, ten R (2+)+i (q) = F (q) q w 0 + w 0 0 (2 i) ( ) l(w) (q w q w w + 0 q w w 0 ) q (2+i ) w 0 +(3 i ) w (3.4) w + w (i ) 2 w (2 i) + 0 ( ) l(w) (q w q w w 0 q w 0 + w 0 0 ) q (5 i ) w 0 +(i ) w (3.5) Remark 3.2. We mentioned in Remark 2.8 tat one way in wic our formulas differ from tose in earlier works is tat ours are unfolded. Now tat we ave stated tis teorem, we can justify tis claim by comparing our formulas (3.4), (3.5) to te corresponding formulas from Teorem 2. in LZ (in te special case k = 2, wic gives te Rogers-Ramanujan series). Eac summand of te series tere corresponds to two terms of our series as given above. In fact, te pairings of terms on even and odd selves (values of j) correspond exactly to te involutions defined in Definitions 2.3, 2.4 respectively. However, tis disparity between even and odd selves is invisible in LZ. Remark 3.3. By replacing some of te expressions in terms of w data wit teir numerical values, we can write te above formulas in a muc more compact form: R 2+i (q) = q (+)(i )+2 (2 i) F (q) ( ) l(w) (q w q w 0)(q w 0 q w ) (q w 0 q w ) q (2+i ) w 0 +(3 i ) w (3.6) R (2+)+i (q) = q (+)2(i )+(+)(2 i) F (q) ( ) l(w) (q w q w 0)(q w 0 q w ) (q w q w 0 ) q (5 i ) w 0 +(i ) w (3.7)

24 7 In te statement of te teorem, we ave cosen to write out te expanded version for several reasons: first of all, tis is te form in wic te exponents actually arise out of te recursions. It also makes it clearer ow te transformations s, f( ) act on tese factors. Finally, in te proof of te Empirical Hypotesis in te next section, tis formulation makes it evident ow cancellations arise in te exponents for te smallest non-zero contribution. However, coosing to write te formulas tis way does necessitate an extra step in te proofs of te edge-matcing, invoking certain identities of te w components in order to properly compare te two different series. In tis section we will point out tese sifts wenever tey occur, but in later sections we will suppress tese reminders. Proof. Te expressions above give two different formulas for te edge cases i.e., i = on self j and i = 2 on self j so we first prove tat tey are compatible. Let R j,i (q) denote te rigt-and side of te formulas above. We will verify tat te difference R j,2 (q) R j+, (q) = 0 We do tis by performing a suitable cange of index on te sum, namely, te two transformations defined above in Definitions 2.3 and 2.4. Te calculation will depend on te parities of te selves involved. Let j = 2 (so tat j + = 2 + ). Ten we ave: R 2,2 (q) R 2+, (q) = F (q) q 2 w0 0 q 2 w + 0 F (q) = F (q) q 3 w0 + w 0 0 ( ) l(w) (q w q w w + 0 q w w 0 ) q (4 ) w 0 +( ) w ( ) l(w) (q w q w w + 0 q w w 0 ) (q w w 0 q w 0 + w 0 0 ) q (4 ) w 0 w ( ) l(w) (q w q w w + 0 q w w 0 ) q (5 ) w 0 w Above, we ave implicitly made te identification w 0 0 = w + 0

25 8 for te leading exponents. We now apply te transformation s (from Definition 2.3) to te Weyl group. Tis is a bijection of te Weyl group, so must leave te above sum invariant. Recall from (2.7) tat te effect of tis transformation on te components w 0 and w : w = w 0 α 0 + w α s w = w 0 α 0 + (2 w 0 w + )α Also recall tat it canges te lengt of eac Weyl element by one. We ave already computed above in Corollary 2.5 te effect of tis transformation on te factors in parenteses. Hence te above sum is transformed into: R 2,2 (q) R 2+, (q) = F (q) q 3 w0 + w 0 0 = F (q) q 3 w0 + w 0 0 s s ( ) l(sw) (q w q w w + 0 q w w 0 ) q (2 w 0 2 w +)+(5 ) w 0 (2 w 0 w +) ( ) l(w) (q w q w w + 0 q w w 0 ) q (5 ) w 0 w Tis final expression is exactly te negation of te starting series. Te anti-commutativity of tis expression proves tat it must be 0. Next, let j = 2 + (so tat j + = 2( + )). Ten we ave: R 2+,2 (q) R 2(+), (q) = F (q) q F (q) q = q 2 w + F (q) w + w w 0 + w ( ) l(w) (q w q w w 0 q w 0 + w 0 0 ) q (3 ) w 0 +( ) w ( ) l(w) (q w q w w 0 q w 0 + w 0 0 ) (q w w + w w 0 q )q (2 ) w 0 +( ) w ( ) l(w) (q w q w w 0 q w 0 + w 0 0 ) q (2 ) w 0 +(2 ) w

26 9 Above, we ave implicitly made te identification for te leading exponents. w w 0 = 0 + w + + w 0 + We now apply to te Weyl group te transformation f, defined in Definition 2.4. Tis is a bijection of te Weyl group, so must leave te above sum invariant. Recall tat te effect of tis transformation is to intercange te components w 0 and w, and tat it leaves invariant te lengt of te Weyl element. Invoking Corollary 2.5, we see tat te above series is transformed into: R 2+,2 (q) R 2(+), (q) = q 2 w + F (q) f(w) W ( ) l(f(w)) (q w q w w 0 q w 0 + w 0 0 ) q (2 ) w +(2 ) w 0 Again te overall effect of tis transformation is to negate te sum, and te anti-commutativity of tis expression proves tat it must be 0. Remark 3.4. Te calculations in tis proof are simpler and more pilosopically satisfying tan te corresponding steps in te proof in LZ (or CKLMQRS for te edge-matcing, wic was suppressed in LZ). Because te series tere were folded (see Remark 3.2), it was necessary in te proofs to break up eac series and reindex different alves separately in order to matc up terms appropriately. Here, te sums are always over te full Weyl group W and terms always matc up naturally. Moreover, te reindexing now comes from a natural transformation of te indexing group W, instead of an ad oc sifting. Now tat we ave verified te edge-matcing, it remains to sow tat tese formulas satisfy te recursion. We andle tis in two cases once more. In bot cases, we take te difference of te two self j series formulas to get te i = 2 entry on self j +.

27 20 First, let j = 2, j + = 2 + : R 2+ (q) R 2+2 (q) = q 2+ q 2+ F (q) q w0 0 F (q) q 2 w0 0 w 0 = F (q) q w0 w 0 (2+) 0 = R (2+)+2 (q) ( ) l(w) (q w q w w + 0 q w w 0 ) q (3 ) w 0 +(2 ) w ( ) l(w) (q w q w w + 0 q w w 0 ) Secondly, let j = 2 +, j + = 2( + ): R (2+)+ (q) R (2+)+2 (q) = q 2+2 q 2+2 F (q) q 2 w + 0 q w w F (q) = q 2 w + 0 F (q) = R (2+2)+2 (q) q (4 ) w 0 +( ) w ( ) l(w) (q w q w w + 0 q w w 0 ) (q w w 0 q w 0 + w 0 0 )q (3 ) w 0 +( ) w ( ) l(w) (q w q w w 0 q w 0 + w 0 0 ) q (4 ) w 0 w ( ) l(w) (q w q w w 0 q w 0 + w 0 0 ) q (3 ) w 0 +( ) w ( ) l(w) (q w q w w 0 q w 0 + w 0 0 ) (q w w + w w 0 q )q (3 ) w 0 w Remark 3.5. From tese calculations, we see tat te most natural way to express te denominators of te recursions (3.3) is as q 2+ = q ( w + + w ) ( w 0 + w 0 ) w w = q +

28 2 wen j = 2 and q 2(+) = q 2 w + 0 w wen j = 2 + (indeed, we ave already implicitly made use of tese expressions to obtain te final lines of te equations above). Remark 3.6. In previous works on motivated proofs, te sum was always reindexed at eac stage of te recursion to ensure tat all summands were nonzero. Tis was natural, because tere was no visible reason for preserving te information of aving terms equal to zero at te beginning of te sum. However, our sums are never reindexed (tey are always over te full Weyl group), so we do end up wit a finite number of summands equal to zero at te start of every R series past te first self. Remark 3.7. It is important to note tat te common factor F (q) plays no role in te proof ere, except for te identification wit te original R i (q), i =,..., k. Te factor F (q) could ave been replaced wit any nonzero formal power series in q, and every step of te proof would ave been identical (beyond te identification wit te original R i (q), i =,..., k), and equivalent to te existing step. However, F (q) is crucial for te Empirical Hypotesis, wic in fact, as we sall see, uniquely determines tis factor. 3.4 Empirical Hypotesis As a consequence of Teorem 3., we are now in a position to formulate and prove te Empirical Hypotesis. Tis is te main ingredient needed to complete te motivated proof. Teorem 3.8 (Empirical Hypotesis). For any j 0 and i =, 2, R j+i (q) = + q j+ γ(q) for some γ(q) Cq. Remark 3.9. Note tat since R j+2 (q) = R (j+)+ (q), Teorem 3.8 implies tat we can write R j+2 (q) = + q j+2 γ(q) were γ(q) is some formal power series.

29 22 Remark 3.0. Te proof of tis teorem is a more involved argument tan for te corresponding result in LZ and CKLMQRS, because it is necessary to translate te data of te series from Weyl notation to integers in order to compare te term of te series to te sape of te proposed Empirical Hypotesis. An overview of te proof: Recall te linear order on W introduced above, in Remark 2.8. In general, for eac R series, te first few summands in tis ordering will be equal to 0. Te first term wic is nonzero will contribute te initial + q j+ in te Empirical Hypotesis, and in general, all oter contributions from it and all subsequent terms will involve only greater powers of q. In te special case i = 2, a closer analysis sows tat te next term after te least nonzero contributor contributes exactly wat is needed to make te series of te form + q j+2. Proof. First, we consider te even-self series, wic from Teorem 3. are of te form R 2+i (q) = 2 F (q) q w0 (i ) 0 w 0 + w 0 0 (2 i) ( ) l(w) (q w q w w + 0 q w w 0 ) q (2+i ) w 0 +(3 i ) w First suppose tat w is one of te first 2 elements (wit respect to te linear order)- i.e., w is eiter of te form w 0 r for r <, or w r for r. It follows from (2.4), (2.5) tat w 0 r 0 w 0 r = r = w r w r 0 (3.8) Considering te corresponding term, we see tat exactly one of tese factors inside te parenteses will be zero in tis case. Specifically, if w = w 0 r for some for r <, ten (q w q w 0 w 0 r 0 + w 0 r ) = (q w q w 0 r ) = 0 (3.9) wile if w = w r for some r, ten (q w 0 q w + w r 0 w r ) = (q w 0 q w r ) = 0 (3.0) We claim tat te first nonzero contribution to te series is from te term w = w 0, and tat tis contribution is of te sape + q 2+ + O(q 2+2 ). Tis claim is justified by te following sequence

30 23 of computations, starting wit te w = w 0 term from te series referenced above: 2 F (q) q w0 (i ) 0 w 0 + w 0 0 ( ) l(w0 ) (q w 0 q w 0 = 2 F (q) q w0 (i ) 0 (2 i) w 0 + w 0 0 0) (q w 0 (2 i) 0 q w 0 + w w 0 ) q (2+i ) w 0 +(3 i ) w 0 0 ( ) {q w 0 ( q w 0 w 0 0 )} { q w 0 ( q w 0 w )} q (2+i ) w 0 +(3 i ) w 0 0 = F (q) ( )2 ( q) ( q 2 ) 2 w 0 (i ) q 0 w 0 + w 0 0 (2 i)+2 w 0 w 0 +(2+i ) w 0 +(3 i ) w In te middle of te calculation, we used te fact tat in te factors q w r for r =,...,, eac r is really w r w r 0. Ten te relations (2.4) - (2.7) allow us to collapse te sum + + into a telescoping sum wic yields te singleton w 0 in te exponent on te next 0 line. Looking more closely at te exponent of q, we can regroup it as 2 ( w i + w 0 ) ( w w 0 ) ( w 0 0 w 0 ) ( w 0 0 w 0 ) + w w 0 ( w 0 0 w 0 ) = = 0 Moreover, expanding F (q) = ( q)( q 2 )( q 3 ) we see tat we can cancel te first 2 factors of F (q) wit te remaining factors in te term. Hence tis term is in fact equal to (q 2, q) = + q 2+ + (as can be seen by using te standard expansion ( q) = + q + q 2 + q 3 + )

31 24 To verify tat all subsequent contributions are O(q 2+ ), we will once again make use of te two bijective transformations of te Weyl group using Corollary 2.5. Te proof ere is essentially by induction: starting wit w 0 (for wic we ave explicitly calculated te contribution), we compute ow te minimal power of q in te next (wit respect to te linear order) term is related to te powers present in te previous term. We will perform two calculations, one for eac of our te transformations (and corresponding respectively to tose w of eac of te two forms w 0 r, w r). For tese calculations, te part of te data tat is te same across all terms (i.e., te leading factors written in terms of F (q) and te components of w 0 ) will not be affected, and ence can be neglected in te computations. First, suppose tat w is of te form w 0 r for some r. Using Definition 2.3 and Corollary 2.5, we are able to express te term corresponding to s w = s w 0 r = w r+ in terms of te w0 r data. Te overall effect is to scale te term by a certain power of q. By combining te factor from Corollary 2.5 and taking te difference of te original and transformed ending exponents, we calculate tat tis power of q is equal to ( ) 2 w 0 2 w + + (2 + i ) w 0 + (3 i )(2 w 0 w + ) (2 + i ) w 0 + (3 i ) w ( ) = 2 w 0 2 w + + (3 i )(2 w 0 2 w + ) = (3 i)(2 w 0 2 w + ) 2 + were for te last inequality olds, we ave used te facts tat i 2 and tat for elements of te form w 0 r wit r, we ave w 0 w. (Notice tat in particular, we get equality in te last line only wen i = 2 and r =, and in tis case te least contribution from tis term is q 2+, wic precisely cancels out te q 2+ from te previous term, giving us a series of te form + q Tis sows te agreement between te edge-matcing and te Empirical Hypotesis). In general, tis confirms tat all iger contributions are O(q 2+ ). Next, suppose tat w = wr for some r +. Here te next Weyl element in te linear order

32 25 is f(w r) = w 0 r. From Definition 2.4 and Corollary 2.5, te relevant parts of te terms tat we need to compare are (q w 0 q w ) q (2+i ) w 0 +(3 i ) w = q (3+i ) w 0 +(3 i ) w q (2+i ) w 0 +(4 i ) w for te w r term and (q w q w 0 )q (2+i ) w +(3 i ) w 0 q (3+i ) w 0 +(3 i ) w q (2+i ) w 0 +(4 i ) w Te ratios between te parts of tese expressions is q ( 2i+) w 0 +(2i ) w for te positive parts and q ( 2i+) w 0 +(2i ) w + for te negative parts. Te ratio between positive parts is te smaller of te two, and we can bound its exponent as (2i ) w w 0 (2i )( + ) Hence te power of q must increase, and ere, too, te contribution is O(q 2+ ). Combining tese results, we ave verified te Empirical Hypotesis for even selves. For odd selves, from Teorem 3., te series are of te form R (2+)+i (q) = F (q) q w + w (i ) 2 w (2 i) + 0 ( ) l(w) (q w q w w 0 q w 0 + w 0 0 ) q (5 i ) w 0 +(i ) w First suppose tat w is one of te first 2 + elements (wit respect to te linear order) - i.e., tat w is eiter of te form w 0 r for r, or w r for r. Ten reasoning as above, it follows from te formulas (3.8), (3.9), and (3.0) tat te terms corresponding to tese elements are 0.

33 26 We claim tat te first nonzero contribution to te series is from te term w = w+, and tat tis contribution is of te sape + q O(q 2+3 ). Tis claim is justified by te following sequence of computations, starting wit te w = w+ term from te series referenced above: F (q) q = F (q) q w + w (i ) 2 w 0 (2 i) + 0 ( ) l(w + ) (q w + q w + 0) (q w + q w + w 0 + w ) q (5 i ) w + +(i ) w 0 + w + w (i ) 2 w 0 (2 i) + 0 ( ) + { q w + 0( q w + w + 0)} { q w + w w 0 ( q )} q (5 i ) w 0 +(i ) w 0 0 = F (q) ( )2+2 ( q) ( q 2+ ) q w + w (i ) 2 w 0 (2 i)+(2+) w 0 w +(5 i ) w 0 +(i ) w As above, te relations (2.4) - (2.7) allow us to collapse te telescoping sum ( + + ) into te singleton w in te exponent on te next line. + 0 Looking more closely at te exponent of q, we can regroup it as 4 w i w + 0 ( w + w ) ( + w 0 + w ) w + + w ( + )( 0 + w + w ) + 0 = ( + ) 2 ( + ) 2 = 0 Moreover, expanding F (q) = ( q)( q 2 )( q 3 ), we see tat we can cancel te first 2 + factors of F (q) wit te remaining factors in te term. Hence tis term is in fact equal to (q 2+, q) = + q To verify tat all subsequent contributions are O(q 2+2 ), we use te same tecnique as for te

34 27 even selves: applying te two transformations to terms coming from te above series and verifying tat exponents always increase. First suppose w is of te form wr for some r +, so tat te next Weyl element in te linear order is f(wr) = wr. 0 By Definition 2.4 and Corollary 2.5, te f( ) transformation acts as negation on te factors in parenteses, and its effect on te ending exponents results in rescaling by a power of q, wose exponent is: (i ) w 0 + (5 i ) w (5 i ) w 0 (i ) w = (2i 6) w 0 + ( 2i + 6) w = 2(3 i) w w wit equality only wen i = 2 and w = w +. Since we ave made te assumption tat te preceding term is at least + O(q 2+2 ), and te minimal exponent ere is at least iger, we conclude tat te contribution ere is O(q 2+2 ). Next suppose w is of te form w 0 r for some r +, so tat te next Weyl element is s w 0 r = wr+. Using Definition 2.3 and Corollary 2.5, te relevant difference in te two corresponding terms is te replacement of (q w q w 0 )q (5 i ) w 0 +(i ) w = q (5 i ) w 0 +(i ) w q (6 i ) w 0 +(i ) w from te w 0 r term wit (q w 0 q w (+) )q (2+)( w 0 w )+(+)+(5 i ) w 0 +(i )(2 w 0 w +) = q (4+i ) w 0 +( i+ ) w q (5+i ) w 0 +( i ) w for te w r+ term. Te ratios of te positive and negative parts ere are respectively q (2i ) w 0 +( 2i+) w +i, q (2i ) w 0 +( 2i+) w +i+

35 28 Te lower ratio is between te positive parts, wit exponent (2i ) w 0 w + i i + (using te fact tat w 0 w + for w in te range we re considering). Terefore it is also true in tis situation tat te power of q increases, and te contribution to te series remains O(q 2+2 ). Having done bot cases, we see tat all iger contributions are O(q 2+2 ), and we ave verified te Empiricial Hypotesis for odd selves. Remark 3.. Our proof of te Empirical Hypotesis provides justification for te linear order we ave defined on W (see Remark 2.8). Also, compared to te proof of te corresponding Empirical Hypotesis in LZ, ere we get additional information as to te contributions to te series from eac term, in terms of its Weyl index. Tis was idden before because of ow te series were written witout leading 0s (see Remark 3.6). 3.5 Combinatorics To complete te motivated proof, we now recall Teorem 2.2 from LZ. Remark 3.2. Te proof we give ere (wic is in te spirit of LZ, Remark 4.) is not te most satisfying pilosopically because it requires us to ave known in advance te sape of te sum sides. In LZ, te primary proof of te teorem is muc more detailed and interesting. In particular, te polynomials i (j) l below are computed explicitly, and te combinatorics of te sum sides are derived directly from just te R series we ave been working wit. However, since our new viewpoint in tis work as noting to add to tis argument, we just give te sorter proof ere for te sake of convenience. Teorem 3.3. For eac j = 0,,..., R j+ (q) is te generating function of partitions in wic subsequent parts differ by at least 2, suc tat te smallest part is greater tan or equal to j +. Proof. Suppose J, J 2,... is an infinite sequence of formal power series in q wic satisfy te recursions (3.3) (wit J in place of R) and te Empirical Hypotesis. By rewriting (3.3) to solve

36 29 for te lowest-indexed series and applying tis formula iteratively, we see tat for eac i =,..., 2, we ave expressions for some polynomials i (j) p J i (q) = 2 p= i (j) p (q)j j+p (q) (q) Cq. Notice tat te coefficients i (j) (q) of tese combinations depend only on te recursions, not directly on te Js. It follows from te Empirical Hypotesis tat te series J,..., J k are uniquely determined (considering te combination at self j determines te first j terms of te series J i just in terms of te i (j) p (q)). Hence, te wole sequence J, J 2,... is uniquely determined. By our work in te earlier sections (te definition of te R i series in (3.3) and Teorem 3.8), te series R i above satisfy tese conditions. Let S i denote te generating functions of te classes of partitions described in te statement of te teorem. By uniqueness, it is now enoug to ceck tat te S i also satisfy te recursions and Empirical Hypotesis. Te Empirical Hypotesis (tat S i = +q i+ + ) follows directly from te definition. (Recall from earlier tat te empty partition of 0 is valid, and vacuously satisfies te conditions). To ceck te recursions, we consider te series S i (q) S i+ (q) q i Te series in te numerator counts partitions satisfying te difference 2 condition, and for wic te smallest part is exactly i. Te denominator as te effect of deleting tis part, and te next smallest part must be no less tan i + 2. Hence tis is te generating function S i+2 (q), and we ave verified te recursions. p

37 30 Capter 4 Gordon-Andrews case - new interpretation 4. Introduction Te Rogers-Ramanujan identities admit a natural generalization, te Gordon-Andrews odd-modulus identities. In explicit power-series form, te identities state tat for any k 2, i =,, k m, m 0, ±(k i+) (mod 2k+) q m = n 0 d k,i (n)q n, (4.) d k,i (n) = te number of partitions of n for wic parts at distance k ave difference at least 2, and appears as a part at most k i times It is clear tat wen k = 2, tese identities reduce to te Rogers-Ramanujan identities. From our perspective, tis generalization comes from a different coice of specialization of te same denominator identity. Ultimately, it corresponds to te graded dimensions of te A () modules at odd level, extending te special case of Rogers-Ramanujan at level 3. As mentioned in te introduction, in LZ, Lepowsky and Zu gave a motivated proof of tese identities. Concepts developed in later works on motivated proofs (in particular, te self picture) were not explicitly present in tis work, but in te appendices to CKLMQRS, te appropriate structure was made explicit. We largely follow tat appendix ere, but use our new viewpoint and notation to igligt te underlying algebraic structure at play. In anticipation of tis section, muc of te work in te Rogers-Ramanujan special case we did out above was written in a form tat allows for straigtforward extension to tis level of generality. Altoug we will not stress tese points in tis capter, remarks analogous to Remarks 3.2, 3.3, 3.4, 3.6, 3.0, and 3. are true in tis setting as well.

38 3 4.2 Definitions and background Fix an integer k 2, and allow te parameter i to range over te values,, k. Consider te following specializations of (2.): e α 0 q k+i, e α q k+ i Tese yield (q k+ i, q 2k+ ) (q k+i, q 5 ) (q 2k+, q 2k+ ) = (q, q) m=,,2k+, m k+i,k+ i,2k+ mod 2k+ = ( ) l(w) q (k+i) w 0 +(k+ i) w (q m, q 2k+ ) Let F (q) denote te series (q, q) = n= (4.2) ( q n ), as above. Again we divide troug tese equations by F (q), and recognize on te product sides of te Gordon-Andrews identities (4.) on te left-and sides. Taking te rigt-and sides, we define a zerot self of series by G i (q) = F (q) ( ) l(w) q (k+i) w 0 +(k+ i) w After defining tis zerot self, we recursively generate iger selves of series according to te following rules: given te k series on te jt self, G (k )j+i (q), i =,, k we tautologically ave G (k )(j+)+ = G (k )j+k and for i = 2,, k, we define G (k )(j+)+i (q) = G (k )j+(k i+)(q) G (k )j+(k i+2) q (j+)(i ) (4.3) We write out tese formulas explicitly for te first self, for expository purposes. For i = 2,, k: G (k )+i (q) = F (q) ( ) l(w) (q w q w 0)q (2k i+) w 0 +(i ) w (4.4) Remark 4.. If we allow i = in te expression (4.4), it stands to reason tat te series we get sould be G (k )+, and ence te same as G k (q) = G (k )0+k from te zerot self. In fact, tis is te case - tis is an example of te edge-matcing penomenon wic will be expounded upon in te next section.

39 32 Let us see ow tis equality comes about in tis instance: te zerot-self formula for G k (q) is G k (q) = F (q) wile te extrapolated first-self formula is ( ) l(w) q 2k w 0 + w G (k )+ (q) (4.5) = ( ) l(w) (q w q w 0)q 2k w 0 F (q) (4.6) = F (q) ( ) l(w) q 2k w 0 + w q (2k+) w 0 (4.7) Te difference between tese two expressions is tus F (q) ( ) l(w) q (2k+) w 0 However, we know tat in te affine Weyl group W, te effect of left-composing wit te fundamental reflection s will not cange te value of w but will cange te lengt by. Hence tis sum is zero because te terms cancel in pairs: w, s w. 4.3 Closed formulas We ave te following closed-form formulas for tese series: Teorem 4.2. Let j 0 and i =,, k. If j = 2 is even, ten (i ) 0 w 0 + w 0 0 G (k )2+i (q) = 2 F (q) q w0 ( ) l(w) (q w q w w + 0 q w w 0 ) q (k+i ) w 0 +(k i+ ) w If j = 2 + is odd, ten G (k )(2+)+i (q) = F (q) q w + w (i )+2 w (k i) ( ) l(w) (q w q w w 0 q w 0 + w 0 0 ) q (2k i+ ) w 0 +(i ) w (k i) (4.8) (4.9)

40 33 Proof. Te expressions above give two different formulas for te edge cases i = k for self j 0 and i = for self j + so we first prove tat tey are compatible. Let G j,i (q) denote te rigt-and side of te formulas above. We will verify tat te difference G j,k (q) G j+, (q) = 0 Tis works in te same way as in te Rogers-Ramanujan case: te involutions s, f( ) (from Definitions 2.3, 2.4) of te Weyl group are applied to te series as canges of basis, and tese ave te effect of negating te difference series. Let j = 2 (so tat j + = 2 + ). Ten we ave: G 2,k (q) G 2+, (q) = F (q) q 2 w0 (k ) 0 ( ) l(w) (q w q w w + 0 q w w 0 ) q 2 w (k ) + 0 F (q) = F (q) q w0 (2k )+ w 0 0 q (2k ) w 0 +( ) w ( ) l(w) (q w q w w + 0 q w w 0 ) (q w w 0 q w 0 + w 0 0 ) q (2k ) w 0 w ( ) l(w) (q w q w w + 0 q w w 0 ) q (2k+ ) w 0 w We now apply te transformation s to te Weyl group. Tis is a bijection of te Weyl group, so must leave te above sum invariant. Using Corollary 2.5, we get: F (q) q w0 (2k )+ w 0 0 s = F (q) q w0 (2k )+ w 0 0 ( ) l(sw) (q w q w w + 0 q w w 0 ) s q (2+)( w 0 w )+(+)+(2k+ ) w 0 (2 w 0 w + ) ( ) l(w) (q w q w w + 0 q w w 0 ) q (2k+ ) w 0 w Tis final expression is exactly te negation of te starting sum. Te anti-commutativity of tis expression proves tat it must be 0.