Affine niltemperley-lieb Algebras and Generalized Weyl Algebras: Combinatorics and Representation Theory

Transcription

1 Affne nltemperley-leb Algebras and Generalzed Weyl Algebras: Combnatorcs and Representaton Theory Dssertaton zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematsch-Naturwssenschaftlchen Fakultät der Rhenschen Fredrch-Wlhelms-Unverstät Bonn vorgelegt von Joanna Menel aus Bonn-Dusdorf Bonn, März 2016

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3 Angefertgt mt Genehmgung der Mathematsch-Naturwssenschaftlchen Fakultät der Rhenschen Fredrch-Wlhelms-Unverstät Bonn 1. Gutachter Prof. Dr. Catharna Stroppel. 2. Gutachter Prof. Dr. Hennng Haahr Andersen Tag der Promoton: Erschenungsjahr: 2016 In der Dssertaton engebunden: Zusammenfassung Lebenslauf

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5 Contents Summary 9 Introducton 11 I Partcle confguratons and crystals 27 I.1 Crystal bases and partcle confguratons 29 I.1.1 Quantum groups and crystal bases of type A n and Ân I Fnte case I Affne case I.1.2 Combnatorcs of partcle confguratons I.2 The affne nltemperley Leb algebra 47 I.2.1 Notaton I.2.2 Related algebras I The affne nlcoxeter algebra I The unversal envelopng algebra of the Le algebra of affne type A 52 I The affne plactc algebra I Combnatoral actons I The creaton/annhlaton algebra I The affne Temperley Leb algebra I.2.3 Gradngs I.2.4 The graphcal representaton of the affne nltemperley Leb algebra.. 59 I.2.5 A normal form of monomals n the affne nltemperley Leb algebra.. 62 I.2.6 Fathfulness of the graphcal representaton I Labellng of bass elements I Descrpton and lnear ndependence of the matrces I.2.7 Projectors I.2.8 Descrpton of the center I.2.9 The affne nltemperley Leb algebra s fntely generated over ts center 83 5

6 Contents I.2.10 An alternatve normal form usng the center I.2.11 Embeddngs of affne nltemperley Leb algebras I.2.12 Classfcaton of smple modules I.2.13 The affne nltemperley Leb algebra s not free over ts center I.2.14 Affne cellularty of the affne nltemperley Leb algebra I.3 The plactc and the partc algebra 99 I.3.1 The classcal and the affne plactc algebra I.3.2 The partc algebra I.3.3 A bass of the partc algebra I.3.4 The acton on bosonc partcle confguratons I.3.5 The center of the partc algebra I.3.6 The affne partc algebra II Generalzed Weyl algebras 119 II.1 A Duflo theorem for a class of generalzed Weyl algebras 121 II.1.1 An overvew of Duflo type theorems II.1.2 Generalzed Weyl algebras and graded modules II Defnton of a GWA and frst observatons II A specal class of GWA s II Weght modules II A characterzaton of hghest weght modules for specal GWA s 126 II Sde remark on generalzed gradngs II.1.3 Descrpton of weght modules n terms of breaks II Gradng of weght modules II Breaks and the submodule lemma II.1.4 Prmtve deals of generalzed Weyl algebras II The man result II The result of [MB98] II The proof of Theorem II.1.4.1: Reducton to weght modules II The proof: The refnement II.1.5 Examples II The frst Weyl algebra II A rank 1 example wth two breaks II A rank 2 example

7 Contents CV 145 Bblography 147 7

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9 Summary Ths thess les at the crossroads of representaton theory and combnatorcs. It s subdvded nto two parts, each of whch s devoted to a partcular combnatoral technque n the study of weght modules. In the frst part, we start out by a short revew of crystal bases for fnte-dmensonal smple modules of the quantum group U q (sl n (C)) and for Krllov Reshetkhn modules of the quantum affne algebra U q ( sl n (C)). We dentfy crystal bases wth combnatorally defned partcle confguratons on a lattce. Such partcle confguratons consst of a fnte number of partcles dstrbuted along a lne segment (the fnte/classcal case) or along a crcle (the affne case). There are two versons present: Fermonc confguratons where only one partcle s allowed at each poston, and bosonc confguratons where arbtrarly many partcles are admssble. Under ths dentfcaton, Kashwara crystal operators correspond to partcle propagaton operators, pushng partcles from one poston n the lattce to another. These operators satsfy the plactc relatons, and we want to descrbe the algebras that act fathfully on the partcle confguratons. It s known that the nltemperley Leb algebra acts fathfully on fermonc partcle confguratons on a lne segment. For bosonc partcle confguratons on lne segments, we prove fathfulness of the acton of the so-called partc algebra, whch we defne as a quotent of the plactc algebra. We construct a bass of the partc algebra, and we descrbe ts center. The queston becomes substantally harder n the affne case. For fermonc partcle confguratons on a crcle t s the affne nltemperley Leb algebra that acts fathfully. Ths s an nfnte dmensonal algebra defned by generators and relatons. Our man results for the affne nltemperley Leb algebras nclude dfferent bases of the algebra, an explct descrpton of ts center, and a classfcaton of ts smple modules. Furthermore, we defne embeddngs of the affne nltemperley Leb algebra on N generators nto the affne nltemperley Leb algebra on N + 1 generators. For bosonc partcle confguratons on a crcle we fnd an nterestng famly of addtonal relatons that are not obvous from the classcal case. 9

10 Summary The second part of the thess exhbts a dfferent combnatoral approach to weght modules, namely that of dscrete geometry appled to the support of a module. Ths tme we consder the representaton theory of generalzed Weyl algebras, a class of algebras that generalzes the defnton of the Weyl algebra, the algebra of dfferental operators on a polynomal rng. Its weght modules allow a beautful descrpton n terms of lattce ponts and hyperplanes. We apply a theorem by Musson and Van den Bergh [MB98] to a specal class of generalzed Weyl algebras, thereby provng a Duflo type theorem statng that the annhlator of any smple module s n fact gven by the annhlator of a smple hghest weght module. 10

11 Introducton The nterplay of representaton theory and combnatorcs bulds on a long tradton. In partcular the study of algebras that admt a noton of hghest weght modules has turned out to be remarkably frutful. Famous examples are provded by unversal envelopng algebras of Le algebras, quantum groups and Weyl algebras. Wthn the usually unfathomable category of all modules over such an algebra, t s the subcategory of weght modules that allows for neat combnatoral descrptons. Weght combnatorcs have been studed extensvely over the past decades, and they contnue to be a source of beautful results wth many applcatons n algebra, geometry and mathematcal physcs. In all of the examples above, the algebra s generated by a nce subalgebra whose representaton theory s well understood e.g. a commutatve subalgebra together wth some addtonal generators that come n pars (often called postve and negatve generators) so that the product or the commutator of each such par les n the nce subalgebra. Weght modules are fully reducble modules over the nce subalgebra, the rreducble summands are called the weght spaces of the module. The labellng set of the somorphsm classes of smple modules over the nce subalgebra s called set of weghts. The postve and negatve generators take weght spaces to weght spaces (or 0) n a controlled way deally, each weght space s taken to one partcular other weght space, so one gets an acton of the postve and negatve generators on the set of weghts. The classcal example s the smple Le algebra sl n (C) wth ts trangular decomposton nto upper and lower trangular matrces and the commutatve subalgebra of dagonal matrces h, together wth ts hghest weght modules n category O wth weghts n h, see [BGG76], [Hum08]. Ths can be generalzed to a theory of Le algebras wth a trangular decomposton as n [MP95, Sectons 2.1, 2.2], [RCW82]. Also the noton of category O can be extended to Le algebras wth a trangular decomposton [Kha15]. Some characterzatons of smple hghest weght modules carry over from the complex semsmple Le algebra case to more general Le algebras wth a trangular decomposton [MZ13]. 11

12 Introducton An mportant result about hghest weght modules for semsmple complex Le algebras s Duflo s theorem [Duf77]. It states that nsde the unversal envelopng algebra, all the annhlators of smple modules are gven by the annhlators of smple hghest weght modules. In contrast, the smple modules themselves are far from beng classfed n general. Ths theorem underlnes the sgnfcance of hghest weght modules nsde the category of all modules over a semsmple complex Le algebra. There are some Duflo type theorems for other famles of algebras known, see Secton II.1.1. One result of ths thess s the proof of a Duflo type theorem for a class of generalzed Weyl algebras. The defnton of weght modules opens many possbltes to apply combnatorcs to representaton theoretc questons. Some of the tools that also appear n ths thess nclude crystal bases, Young tableaux, and geometry of weght lattces. But also further combnatoral technques lke gradngs, central characters, dagrammatcal calculus, and (affne) cellular structures are present. Certan crystal bases for hghest weght modules of the quantum group U q (sl n (C)) and the quantum affne algebra U q ( sl n (C)) can be dentfed wth partcle confguratons on a lattce, so that the Kashwara operators correspond to partcle propagaton operators. Such partcle confguratons were used n [KS10, Theorem 1.3] to descrbe the ŝl n (C)-Verlnde algebras, whch n turn can be dentfed wth a quotent of the quantum cohomology rng of the Grassmannan, see e.g. [Buc03], [Pos05] and see [ST97] for a presentaton by generators and relatons. An alternatve combnatoral realsaton n terms of vcous and osculatng walkers s gven e.g. n [Kor14]. 12

13 Introducton Overvew of the thess The thess conssts of two parts. The frst part on Partcle confguratons and crystals s splt nto three chapters, the second part on Generalzed Weyl algebras contans a sngle chapter. All chapters are ndependent from each other, although we nclude cross-references to ndcate connectons among them. Conventons and notaton In both parts of the thess we use the followng conventons unless stated otherwse: By a module, we always mean a left module. All rngs and algebras are assocatve and untal. We denote our ground feld by k. If the ground feld should satsfy any addtonal propertes (uncountable, algebracally closed and/or of characterstc 0) we ndcate ths n the begnnng of the chapter or secton where t apples. In Chapter I.1 we work over the complex numbers k = C. In most of Chapters I.2 and I.3 t suffces to assume that k be a (commutatve) rng, for detals see Remark I We use δ to denote the Kronecker symbol,.e. δ xy = 1 f x = y and δ xy = 0 f x y. The symmetrc group generated by m 1 smple transpostons (, + 1) s denoted by S m. Part I: Partcle confguratons and crystals The frst part of the thess deals wth the (classcal and affne) plactc algebra, and two nterestng quotents: The affne nltemperley Leb algebra, a quotent of the affne local plactc algebra that has been known before, and the partc algebra, a quotent of the classcal local plactc algebra that we ntroduce n ths thess. These algebras are defned by generators and relatons over the ground feld (or ground rng) k, and they appear n the study of representaton theory and crystal combnatorcs of U(sl n (C)) and U( sl n (C)). Let us brefly ntroduce these algebras and explan where they come from and why they are nterestng. After that we gve an overvew of our results. For precse defntons and statements see the cross-references. 13

14 Introducton The classcal (local) plactc algebra s generated by a 1,..., a N 1 subject to the so-called plactc relatons a a j = a j a for j > 1, a a 1 a = a a a 1 for 2 N 1, a a +1 a = a +1 a a for 1 N 2. For the affne verson of the plactc algebra, take generators a 1,..., a N 1, a 0 wth the same relatons, except that the ndces of the generators are now read modulo N. In partcular we have addtonal relatons a 0 a N 1 a 0 = a 0 a 0 a N 1 and a 0 a N 1 a N 1 = a N 1 a 0 a N 1, and the generators a 0 and a N 1 are neghbours that do not commute (Defnton I.3.1.2). The classcal plactc algebra was studed n [FG98]. It s a quotent of the algebra over the monode plaxque defned by Lascoux and Schützenberger [LS81]. These relatons are also known as 0-Serre relatons from a specalsaton of the negatve or postve half of U q (sl N (C)) to q = 0 (Remark I.1.1.9), and they are precsely the relatons satsfed n the Hall monod from [Re01], [Re02] (classcal type A) and [DD05] (affne type Â). Moreover, the Kashwara operators on certan crystals of type A and  satsfy the above relatons (Secton I.1.1). These are the crystals B(ω k ) and B(kω 1 ) assocated wth the alternatng representaton Λ k (C N ) and the symmetrc representaton Sym k (C N ) of sl N (C), and n the affne case the correspondng Krllov-Reshetkhn crystals, as dscussed n Chapter I.1. In [KS10], the plactc algebra appears n the study of certan partcle confguratons. Ths s also our pont of vew n Chapter I.2 and I.3: Combnatorally, a partcle confguraton s defned as a tuple (k 1,..., k N 1, k 0 ) n Z N 0 (called bosonc) or n {0, 1}N (fermonc). One can thnk of such a tuple as a fnte number of partcles dstrbuted on a dscrete lattce of N postons on a lne segment (the fnte/classcal case) or along a crcle (the affne case). In bosonc confguratons, arbtrarly many partcles are admssble, whle n fermonc confguratons at most one partcle s allowed at each poston Example for N = 8: A bosonc partcle confguraton on a lne segment and a fermonc partcle confguraton on a crcle. 14

15 Introducton The generators a act on the partcle confguratons (or ther k-span) by lowerng k by 1 and ncreasng k +1 by 1, f possble. If not possble,.e. because k = 0 or, n the fermonc case, k +1 = 1, the result s 0. In the pcture ths would correspond to (clockwse) propagaton of a partcle from poston to + 1 (Sectons I.2.4 and I.3.4). Ths acton can be dentfed wth the acton of Kashwara operators f on crystals B(ω k ) and B(kω 1 ) (Secton I.1.2). On affne partcle confguratons, the addtonal generator a 0 takes a partcle from poston 0 and moves t to poston 1. If we consder the k[q]-span nstead of the k-span, we can keep track of the applcaton of a 0 by multplcaton wth an addtonal factor q (bosonc) or ±q (fermonc). In Chapter I.2 we descrbe a quotent of the affne plactc algebra that acts fathfully on the k[q]-span of fermonc partcle confguratons on a crcle. Ths s the affne nltemperley Leb algebra n TL N : It s defned by the addtonal nl relaton a 2 = 0 for all. Together wth the plactc relatons we obtan mmedately that also a a ±1 a = 0 for all, where we take the ndces modulo N. The subalgebra of n TL N generated by a 1,..., a N 1 s the (classcal/fnte) nltemperley Leb algebra ntl N. Chapter I.3 s devoted to the quotent of the classcal plactc algebra that acts fathfully on the k-span of the bosonc partcle confguratons on a lne segment. The addtonal defnng relaton s a a 1 a +1 a = a +1 a a 1 a for all 2 N 2. We call ths the partc algebra because of ts fathful acton on the partcle confguratons. The correspondng acton of the affne plactc algebra on bosonc partcle confguratons on a crcle s much harder to descrbe: We encounter an nfnte famly of addtonal relatons of the form a m +1a m a m 2a m 1a 2m a m +1a m a m 2a m 1 = a m j+1a m j+2... a m j 2a m j 1a 2m j a m j+1a m j+2... a m j 2a m j 1 for all, j Z/NZ, m Z 1, and t s not yet clear whether these relatons together wth a a 1 a +1 a = a +1 a a 1 a for all Z/NZ suffce to produce a fathful acton. 15

16 Introducton The followng pcture recaps the relatonshp among the algebras studed n Part I: (affne) plactc algebra a a j = a j a f j > 1 a a 1 a = a 2 a 1 a a +1 a = a +1 a 2 a 2 = 0 a +1a 2 a 1 = a a 1 a +1 a (affne) nltemperley Leb algebra a a j = a j a f j > 1 a a 1 a = 0 a a +1 a = 0 a 2 = 0 acts on (affne) partc algebra a a j = a j a f j > 1 a a 1 a = a 2 a 1 a a +1 a = a +1 a 2 a +1 a 2 a 1 = a a 1 a +1 a acts on classcal and affne versons of fermonc partcle confguratons/ crystal B(ω k ) classcal and affne versons of bosonc partcle confguratons/ crystal B(kω 1 ) 16

17 Introducton Chapter I.1: Crystal bases and partcle confguratons The frst chapter s manly devoted to a revew of crystals of classcal type A and affne type Â. We brefly recall the basc defntons of quantum groups and quantum affne algebras, ther fnte dmensonal rreducble modules, and ther crystal bases n Secton I.1.1. We consder the acton of Kashwara operators on the crystals B(ω k ) and B(kω 1 ) for the smple U q (sl N (C))-modules L q (ω k ) and L q (kω 1 ), correspondng to the alternatng representaton Λ k (C N ) and the symmetrc representaton Sym k (C N ) of U(sl N (C)), respectvely. In affne type  we study the crystals of the Krllov-Reshetkhn modules W k,1 and W 1,k that are somorphc to L q (ω k ) and L q (kω 1 ) as U q (sl N (C))-modules, respectvely. In ths specal case t s partcularly easy to descrbe ths operaton. We make the followng two observatons for classcal type A, as well as the analogous observatons for Krllov Reshetkhn crystals n affne type Â: ˆ On B(ω k ) and B(kω 1 ), the Kashwara operators satsfy the plactc relatons,.e. the 0-Serre relatons. ˆ The crystals B(ω k ) and B(kω 1 ) can be dentfed wth fermonc and bosonc partcle confguratons, so that the acton of the Kashwara operators s dentfed wth partcle propagaton operators. These fermonc and bosonc partcle confguratons are defned purely combnatorally n Secton I.1.2. Chapter I.2: Affne nltemperley Leb algebras The man result of ths chapter s a descrpton of the center of the affne nltemperley Leb algebra n TL N over any ground feld. Only two tools are used: a fne gradng on n TL N and a fathful representaton of n TL N on fermonc partcle confguratons on a crcle. We gve another, more drect proof of the fathfulness result from [KS10, Proposton 9.1] by constructng a bass for n TL N that s especally adapted to the problem. Ths bass has further advantages: It can be used to prove that the affne nltemperley Leb algebra s fntely generated over ts center. Hence, central quotents are fnte dmensonal. Also, t can be used to exhbt an explct embeddng of n TL N nto n TL N+1 defned on bass elements that otherwse would not be apparent, snce the defnng relatons of these algebras are affne, and there s no embeddng of the correspondng Coxeter graphs. As mentoned above, the affne nltemperley Leb algebra n TL N acts fathfully on fermonc partcle confguratons on a crcle. Ths s the graphcal representaton from [KS10] (see also [Pos05]), whch we use n our descrpton of the center of n TL N. We 17

18 Introducton consder crcular partcle confguraton havng N postons, where k N partcles are dstrbuted among the postons on the crcle so that there s at most one partcle at each poston. On the space span k[q] {fermonc partcle confguratons of k partcles on a crcle wth N postons}, the generators a of n TL N act by sendng a partcle lyng at poston to poston + 1. Addtonally, the partcle confguraton s multpled by ±q when applyng a 0. The precse defnton s gven n Secton I.2.4, here s a pcture that llustrates the acton: Example for N = 8: Applcaton of a 3 a 2 a 5 to the partcle confguraton (0, 1, 2, 5) gves (0, 1, 4, 6). We proceed as follows: In Secton I.2.1, we ntroduce our notaton. In Secton I.2.2 we explan the connecton between affne nltemperley Leb algebras and many other algebras, such as the affne plactc algebra and the affne Temperley Leb algebra, and we brefly recall the relatonshp wth the small quantum cohomology rng of the Grassmannan. The Z N -gradng of n TL N s gven n Secton I.2.3, and ts mportance for the descrpton of the center s dscussed. In Secton I.2.4, we gve a detaled defnton of the n TL N -acton on fermonc partcle confguratons on a crcle. Theorem I of that secton recalls [KS10, Proposton 9.1] statng that the representaton s fathful. In [KS10], ths fact s deduced from the fnte nltemperley Leb algebra case, as treated n [BJS93] and [BFZ96, Proposton 2.4.1]. We gve a complete, self-contaned proof n Secton I.2.6. Our proof s elementary and reles on the constructon of a bass n Secton I.2.5. We use a normal form algorthm that reorders the factors of a nonzero monomal. Our bass s remnscent of the Jones normal form for reduced expressons of monomals n the Temperley Leb algebra, as dscussed n [RSA14], and s charactersed n Theorem I as follows (see also Theorem I whch gves a dfferent descrpton): Theorem (Normal form). Every nonzero monomal n the generators a j of n TL N can be rewrtten unquely n the form (a (m) 1... a (m) k )... (a (n+1) 1... a (n+1) k )(a (n) 1... a (n) k )... (a (1) 1... a (1) k )(a 1... a k ) 18

19 Introducton wth a (n) l {1, a 0, a 1,..., a N 1 } for all 1 n m, 1 l k, such that a (n+1) l {1} f a (n) l = 1, {1, a j+1 } f a (n) l = a j. The factors a 1,..., a k are determned by the property that the generator a l 1 does not appear to the rght of a l n the orgnal presentaton of the monomal. Alternatvely, every nonzero monomal s unquely determned by the followng data from ts acton on the graphcal representaton: ˆ the nput partcle confguraton wth the mnmal number of partcles on whch t acts nontrvally, ˆ the correspondng output partcle confguraton, ˆ the power of q by whch t acts. For the proof of ths result, we recall a charactersaton of the nonzero monomals n n TL N from [Gre02]. Al Harbat [Alh13] has recently descrbed a normal form for fully commutatve elements of the affne Temperley Leb algebra, whch dffers from ours when passng to n TL N. In Secton I.2.7 we defne specal monomals that serve as the projectons onto a sngle partcle confguraton (up to multplcaton by ±q). Based on ths, n Secton I.2.8 we state the man result (Theorem I.2.8.5) of the chapter: Theorem. The center of n TL N s the subalgebra C N = Cent(n TL N ) = 1, t 1,..., t N 1 k[t 1,..., t N 1 ] (t k t l k l), where the generator t k = ( 1) k 1 a(î) s the sum of monomals a(î) correspondng I =k to partcle confguratons gven by ncreasng sequences = {1 1 <... < k N} of length k. The monomal a(î) sends partcle confguratons wth n k partcles to 0 and acts on a partcle confguraton wth k partcles by projectng onto and multplyng by ( 1) k 1 q. Hence, t k acts as multplcaton by q on the confguratons wth k partcles. Our N 1 central generators t k are essentally the N 1 central elements constructed by Postnkov. Lemma 9.4 of [Pos05] gves an alternatve descrpton of t k as product of the k-th elementary symmetrc polynomal (wth factors cyclcally ordered) wth the (N k)-th complete homogeneous symmetrc polynomal (wth factors reverse cyclcally ordered) n the noncommutng generators of n TL N. The above theorem shows that n 19

20 Introducton fact these elements generate the entre center of n TL N. In Secton I.2.9, we establsh that n TL N s fntely generated over ts center. In Secton I.2.10 we descrbe an alternatve normal form for monomals n n TL N usng the generators t k of the center. Usng the fathfulness of the graphcal representaton, we defne monomals e j that move partcles from postons j = {1 j 1 <... < j k N} to = {1 1 <... < k N} so that the power of q n ths acton s mnmal. Then the man result s Theorem I : Theorem. The set of monomals {1} {t l k e j l Z 0, 1 = j = k N 1, 1 k N 1} defnes a k-bass of the affne nltemperley Leb algebra n TL N. In Secton I.2.11, we defne yet another monomal bass for n TL N ndexed by pars of partcle confguratons together wth a natural number ndcatng how often the partcles have been moved around the crcle. Wth ths bass at hand, we obtan nclusons n TL N n TL N+1. The nclusons are not as obvous as those for the nlcoxeter algebra nc N havng underlyng Coxeter graph of type A N 1, snce one cannot deduce them from embeddngs of the affne Coxeter graphs. Our result, Theorem I , reads as follows: Theorem. For all 0 m N 1, there are untal algebra embeddngs ε m n TL N n TL N+1 gven by a a for 0 m 1, a m a m+1 a m, a a +1 for m + 1 N 1. In Secton I.2.12 we turn towards the representaton theory of n TL N : In ths secton and the remander of Chapter I.2 we have to assume that the ground feld k of n TL N s restrcted to be an uncountable algebracally closed feld (of arbtrary characterstc). Let χ be an algebra homomorphsm C N k. Then wth the help of localsatons wth respect to central elements, we classfy the smple modules over n TL N character χ n Theorem I as follows. wth central Theorem. Up to somorphsm, there s precsely one smple module of n TL N wth central character χ. The smple modules of n TL N are gven up to somorphsm by ) the trval onedmensonal module k wth trval central character, ) the ( N k )-dmensonal module k k N wth central character χ(t k ) k {0}, χ(t l ) = 0 for all l k. 20

21 Introducton The localsaton wth respect to multplcatve subsets of the center can be consdered as pseudo-commutatve localsaton snce the Ore condtons are for free. In Secton I.2.13 we use these localsatons together wth a rank argument to show that n TL N s not free over ts center. In analogy to the affne Temperley Leb algebra one would expect that also the affne nltemperley Leb algebra can easly be equpped wth the structure of an affne cellular algebra n the sense of [KX12]. Then the classfcaton of smple modules for n TL N would follow from the general approach for affne cellular algebras. However, affne cellularty does not pass n an obvous way to the nl-case. In Secton I.2.14 we dscuss three approaches to dentfy n TL N as an affne cellular algebra. Chapter I.3: The plactc and the partc algebra Analogous to the results for the affne nltemperley Leb algebra n Chapter I.2, our man results n ths chapter are a descrpton of the center of the partc algebra and the constructon of a bass. Usng ths bass we prove that the acton of the partc algebra on bosonc partcle confguratons s fathful. Agan here s a pcture llustratng ths acton: Example for N = 9: The partcle confguraton (3, 0, 0, 1, 0, 1, 2, 0, 1), and the element a 6 a 5 a 4 actng on t. In Secton I.3.1 we recall the defnton of the classcal and affne plactc algebra, and we put t nto the context of the exstng lterature. Frst we study the classcal plactc algebra: In Secton I.3.2 we dscuss an acton on bosonc partcle confguratons on lne segments, and we defne the quotent of the classcal plactc algebra named partc algebra by the addtonal relaton a a 1 a +1 a = a +1 a a 1 a for 2 N 2. 21

22 Introducton Snce these relatons only nvolve permutatons of the generators we can defne two gradngs on the partc algebra, by the word length and by how often each generator occurs, smlar to the affne nltemperley Leb algebra before. In Secton I.3.3 we construct a normal form of the monomals n the partc algebra. Our man result of ths secton s Theorem I Theorem. The partc algebra P part N has a k-bass gven by monomals of the form {a d N 1 N 1... ad 2 2 ak 1 1 ak ak N 1 N 1 d d 1 + k 1 for all 3 N 1, d 2 k 1 } where d, k Z 0 for all 1 N 1. In Secton I.3.4 we consder the acton of the classcal plactc and the partc algebra on bosonc partcle confguratons, and we obtan the followng fathfulness result n Theorem I Theorem. The acton of the partc algebra P part N fathful. on bosonc partcle confguratons s Ths allows us to defne a labellng of the monomals n normal form. We get an alternatve descrpton of the bass from Theorem I n Proposton I If we wrte a j = a d N 1 N 1... ad 2 2 ak 1 1 ak ak N 1 N 1, t can be reformulated as follows: Theorem. The set of monomals {1} {a j j = (k 1, k 2, k 3..., k N 1, 0), = (0, k 1 d 2, k 2 + d 2 d 3,..., k N 1 + d N 1 )} wth k 1,..., k N 1 Z 0 and d d 1 + k 1 Z 0 for all 3 N 1, d 2 k 1, defnes a k-bass of the partc algebra. In Secton I.3.5 we descrbe the center of the partc algebra: Theorem. The center of the partc algebra P part N {a r N 1a r N 2... a r 2a r 1 r 0}. s gven by the k-span of the elements Fnally, n Secton I.3.6 we turn to the affne case. We defne the affne partc algebra and we consder ts acton on affne bosonc partcle confguratons. Ths s substantally harder to understand than the classcal case, n partcular we fnd a new type of relatons of the form a m +1a m a m 2a m 1a 2m a m +1a m a m 2a m 1 = a m j+1a m j+2... a m j 2a m j 1a 2m j a m j+1a m j+2... a m j 2a m j 1 for all, j Z/NZ, m Z 1. 22

23 Introducton We have not yet found a nce normal form for monomals for the affne partc algebra (and nether for ts quotent wth respect to the new type of relatons). For the constructon of the normal forms of the partc algebra and the affne nltemperley Leb algebra, t was helpful to know the fathful representatons on partcle confguratons. The partcle confguratons could be used for labellng sets of the bass elements. Ths approach fals for the affne partc algebra snce t does not act fathfully on affne bosonc partcle confguratons. It s unclear whether fathfulness holds for the quotent wth respect to the new type of relatons. Part II: Generalzed Weyl algebras Generalzed Weyl algebras (GWA s) were ntroduced by Bavula n [Bav92]. A GWA s defned over a untal assocatve commutatve k-algebra R that s a noetheran doman, where k s an algebracally closed ground feld of characterstc 0. For any choce of n nonzero elements t = (t 1,..., t n ) n R and n parwse commutng algebra automorphsms σ = (σ 1,..., σ n ) n Aut(R) such that σ (t j ) = t j for all j the correspondng GWA A = R(σ, t) s the k-algebra generated over R by 2n addtonal generators X, Y, 1 n, wth relatons X r = σ (r)x, X Y = σ (t ), [X, X j ] = 0, Y r = σ 1 (r)y, Y X = t, [Y, Y j ] = 0, [X, Y j ] = 0 for all 1, j n wth j and all r R. It s a Z n -graded algebra wth deg(x ) = e and deg(y ) = e where we denote by e the -th standard bass vector of Z n. Chapter II.1: Duflo Theorem for a Class of Generalzed Weyl Algebras The man result of ths chapter s a Duflo type theorem for a class of generalzed Weyl algebras (GWA s). For the unversal envelopng algebra of a semsmple Le algebra over k, Duflo s Theorem [Duf77] states that all ts prmtve deals (.e. the annhlators of smple modules) are gven by the annhlators of smple hghest weght modules. In contrast, the smple modules themselves are far from beng classfed n general. Now t s possble to defne hghest weght modules for GWA s and therefore natural to ask whether an analogous statement holds. We prove a Duflo type theorem for a specal 23

24 Introducton class of GWA s usng a theorem by [MB98] that relates the annhlator of a smple weght module to ts support. Ths chapter s subdvded as follows: In Secton II.1.1 we provde a quck overvew of Duflo type theorems. In Secton II.1.2 we revew generalzed Weyl algebras, and we ntroduce our specal class of GWA s. In partcular, our base rng s always a polynomal rng R = k[t 1,..., T n ] and the automorphsms are gven by translatons σ (T j ) = T j δ j b as consdered already n [Bav92]. We dscuss hghest weght modules and graded modules over generalzed Weyl algebras. We characterze moreover the hghest weght modules as those modules wth a locally nlpotent acton of the X. In Secton II.1.3 we prepare to apply the result from [MB98] to our class of GWA s: We recall the descrpton of weght modules by ther support whch s gven n terms of lattce ponts and hyperplanes from [Bav92]. These hyperplanes break the weght lattce nto regons, and a weght module can be charactersed by these regons and ts defnng breaks. Ths s made precse n Defnton II We gve a careful descrpton of the break condtons. In Secton II.1.4 we formulate and prove the man theorem of the chapter: Theorem. Let A = R(σ, t) be a GWA of rank n as defned n Secton II.1.2 where we assume R = k[t 1,..., T n ], σ (T j ) = T j δ j b for b k {0} and t k[t ] k[t 1,..., T n ], t k. Then all prmtve deals of A,.e. the annhlator deals of smple A-modules, are gven by the annhlators of smple hghest weght A-modules L(m) of hghest weght m mspec(r). The man tool s the Duflo type theorem from [MB98]. We show t apples to our stuaton and mprove t by showng that t s enough to consder the much smaller class of hghest weght modules (as n the classcal Duflo theorem). We provde a lst of mportant examples of GWA s to whch the man theorem apples, e.g. central quotents of the unversal envelopng algebra U(sl 2 (C)) and ts generalsatons by [Sm90] as dscussed n [Bav92, Example 1.2.(4)]. We nclude a dscusson why we requre our assumptons on the specal class of GWA s. In Secton II.1.5 we conclude the chapter by some examples that llustrate the relatonshp between the annhlator and the support of smple hghest weght modules. 24

25 Introducton Publcatons and Coauthorshps Parts of ths thess have been publshed or accepted for publcaton durng the PhD project: Most of Chapter I.2 as well as the correspondng parts of ths ntroducton can be found n the paper [BM16] wth Georga Benkart. Except for Lemma II.1.2.2, all of Chapter II.1 s publshed n [Me15]. Sectons I.2.12 and I.2.13 grew out of dscussons wth Gwyn Bellamy and Ul Krähmer. Acknowledgements I am deeply thankful to my advsor Catharna Stroppel for sharng her deep nsghts and her enthusasm, for patently standng me by and supportng me. It has been a huge pleasure to work under her gudance, and I enjoyed every sngle hour durng our many, many dscussons. From the frst tme that I heard about Le algebras to our most recent meetng she has been an nsprng teacher and a fantastc mentor, and I am grateful for all the tme we spent together n great workng atmosphere. Durng my PhD project I spent fve months at QGM Aarhus and fve more months at the Unversty of Uppsala. I would lke to thank Hennng Haahr Andersen and Volodymyr Mazorchuk for generously hostng me, for gvng me ther tme and for the numerous nterestng dscussons that we had. I enjoyed my stays n Aarhus and n Uppsala a lot, and I would lke to thank the members of QGM Aarhus and the people at the Department for Mathematcs n Uppsala for ther knd hosptalty. I am grateful to my coauthor Georga Benkart for our collaboraton, ncludng long dscussons and exctng example computatons. I thank the MSRI Berkeley for gvng us the opportunty to start ths collaboraton durng the programme on Noncommutatve Algebrac Geometry and Representaton Theory. I would lke to thank Gwyn Bellamy, Kenneth Brown, Chrstan Korff and Ul Krähmer for knd advce and dscussons about the affne nltemperley Leb algebra durng a short vst to the Unversty of Glasgow, and I am grateful to Jonas Hartwg for nterestng conversatons about GWA s. I heartly thank the members of the representaton theory workng group n Bonn for sharng ther knowledge, gvng me support and advce. I am partcularly ndebted to Hanno Becker, Mchael Ehrg, Denz Kus, and Danel Tubbenhauer for lots of feedback and dscussons about ths thess, for proofreadng, and for bearng wth my terrble 25

26 Introducton puns. I thank Vktorya Ozornova for comments on the thess, and further thanks are due to the referees for many mprovements of the two artcles underlyng ths thess. I gratefully acknowledge the support by the IMPRS programme of the Max Planck Insttute for Mathematcs, the Hausdorff stpend of the Bonn Internatonal Graduate School, and the scholarshp of the Deutsche Telekom Stftung. Ther generous fundng and the extremely helpful admnstratve staff enabled me to carry out my work, and I learned a lot from the exchange wth the people I met thanks to these programmes. My frends at the Mathematcal Insttutes n Bonn, Aarhus and Uppsala made my lfe as a PhD student very pleasant. The lst of reasons s long, and t ncludes many mathematcal dscussons, feedback, and recprocal encouragement as well as common coffee breaks, QGM lounge meetngs, fka sessons, tea sessons, sush dnners, Vetnamese dnners, cookng together n the tny ktchen of the Mathematcal Insttute n Bonn, bakng delcous cookes, processng tons of chestnuts, hkng n Calforna, Catalona, Corsca, Scotland and the Sebengebrge, cnema vsts, shared rooms n strange hotels, shared offces, and many non-mathematcal dscussons. I am lucky to have a wonderful famly and patent frends who supported me uncondtonally durng the past years, even when I was fully absorbed by my thess. It s a pleasure to thank them for gvng me so much of ther tme and energy! 26

27 Part I. Partcle confguratons and crystals 27

28

29 I.1. Crystal bases and partcle confguratons In ths chapter we dscuss the relatonshp of partcle confguratons on a lattce wth crystal combnatorcs n type A and Â. It can be seen as a motvaton for the defntons of the affne nltemperley-leb algebra, the plactc and the partc algebra that we dscuss n the followng chapters. Ths chapter s otherwse ndependent of the followng chapters. In Secton I.1.1 we revew crystal bases for the quantum group U q (sl n (C)) and the quantum affne algebra U q ( sl n (C)), and we dscuss relatons among Kashwara operators. In Secton I.1.2 we descrbe partcle confguratons followng [KS10] and we dscuss dentfcatons of crystal and partcle combnatorcs. Throughout the chapter we work over the complex numbers k = C for convenence. For tensor products over C we wrte nstead of C. We wrte C(q) for the feld of ratonal functons n the varable q. I.1.1. Quantum groups and crystal bases of type A n and Ân In ths secton we revew crystal bases for the quantum groups U q (sl n (C)) and U q ( sl n (C)) and fx our notaton. We follow manly [HK02] and [Jan96] unless otherwse stated. We focus on type A n and Ân, for more general statements see the references. I Fnte case Let sl n (C) be the Le algebra of traceless complex n n-matrces wth standard Cartan subalgebra h consstng of the dagonal matrces generated by h = e e (+1)(+1) for 1 n 1. Here e denotes the elementary matrx where the (, )th entry s one and all other entres are zero. The root decomposton of sl n (C) wth respect to the adjont h- acton s gven by sl n (C) = α Φ sl n (C) α and smple roots α = ε ε +1 h. Here ε denotes 29

30 I.1. Crystal bases and partcle confguratons the functon on h that returns the th dagonal entry, and Φ = span Z {α 1,..., α n 1 } s the root lattce of sl n (C). In our notaton we do not dstngush between lnear functons on h and lnear functons on the dagonal matrces. The fundamental weghts are gven by ω = ε ε. We denote the weght lattce by P = span Z {ω 1,..., ω n 1 }. It contans the domnant ntegral weghts P + = span Z 0 {ω 1,..., ω n 1 }. The fnte dmensonal smple sl n (C)-modules L(λ) are labelled by ther domnant ntegral hghest weghts λ P + h = span C {ε 1 n}/span C {ε ε n }. Such a domnant ntegral hghest weght can be represented by an element of the form λ = λ 1 ε λ n 1 ε n 1 wth coeffcents λ 1... λ n 1 Z 0. Ths n turn s dentfed wth parttons (λ 1,..., λ n 1 ) wth n 1 rows of length λ. Now we turn to the quantum group: I Defnton. The quantum group U q (sl n (C)) s the untal assocatve C(q)- algebra generated by formal generators E, F, K ±1 for 1 n 1 wth relatons K K 1 E 2 E ±1 [2] q E E ±1 E + E ±1 E 2 = 0, F 2 F ±1 [2] q F F ±1 F + F ±1 F 2 = 0, = 1 = K 1 K for 1 n 1, K j E = q α (h j ) E K j for 1, j n 1, K j F = q α (h j ) F K j for 1, j n 1, K K 1 [E, F j ] = δ j q q 1 for 1, j n 1, [E, E j ] = 0 for j > 1, [F, F j ] = 0 for j > 1, where [n] q = qn q n s the usual notaton for quantum ntegers, so [2] q q 1 q = q + q 1. It can be equpped wth a Hopf algebra structure where n partcular the comultplcaton appled to F s gven by (F ) = F 1 + K F, the comultplcaton appled to E s (E ) = E K E, and the elements K ±1 are grouplke, for 1 n 1. I Remark. Ths s the adjont form of U q (sl n (C)) n the sense of [BG02], where the generators K correspond to the generators α of the root lattce Φ of sl n (C). Alternatve forms of the quantum group U q (sl n (C)) can be defned for the (fner) weght lattce or any other lattce lyng n between those two, see [BG02, Secton 1.6.3], [CP95a, Secton 9.1.A]. Furthermore, there s the Drnfeld-Jmbo quantum algebra whose elements are formal power seres n e, f and h over the feld C[[h]], see [CP95a, Defnton 6.5.1], [Kas95]. There s a map of Hopf algebras from the quantum group defned above nto 30

31 I.1.1. Quantum groups and crystal bases of type A n and Ân the Drnfeld-Jmbo quantum group by q e h 2, K ±1 e ± h 2 h, F e h 4 f and E e h 4 e, see [Kas95, Proposton XVII.4.1] for n = 2. We are only nterested n weght modules,.e. U q (sl n (C))-modules wth a weght space decomposton wth respect to the acton of K, 1 n 1, so that the K act by scalars n C(q) on the weght spaces. In partcular, we consder weght modules wth weghts of the form ±q µ for µ P h, meanng that K acts by ±q µ(h ), for all 1 n 1. All fnte dmensonal U q (sl n (C))-modules are completely reducble nto smple hghest weght modules of hghest weght ±q λ wth λ P +, see [CP95a, Propostons , ]. In other words, the fnte dmensonal hghest weght modules are labelled by parttons λ together wth a choce of (n 1) sgns, so that K acts by ±q λ, for all 1 n 1. One usually prefers the choce of all sgns equal to +1 snce the subcategory of these so-called type 1 modules s closed under tensor products. The abelan subcategory of fnte dmensonal U q (sl n (C))-modules wth a fxed choce of sgns s equvalent to the abelan category of fnte dmensonal sl n (C)-modules. For type 1, ths s an equvalence of monodal categores. Under ths equvalence, the fnte dmensonal smple sl n (C)-module L(λ) s mapped to the smple U q (sl n (C))-module L q (λ) of type 1 wth the same character, see [BG02, Secton I.6.12]. Here and n the followng we adopt the shorthand notaton of wrtng λ for +q λ. Let us now recall the combnatorcs of some specal crystals for sl n (C). We do not ntroduce Kashwara operators and crystal bases n detal. We refer to [Kas91], but also e.g. to [HK02, Secton 4] for the general statements and background materal and to [HK02, Sectons 7.4, 8.2] for detals about type A n. Let f denote the Kashwara operator on a U q (sl n (C))-module M assocated wth the operator F U q (sl n (C)),.e. f u = k F (k+1) u k for a weght vector u M µ wrtten n the form u = k F (k) u k wth u k M λ+kα ker(e ). Here F (k) = 1 [k] F k q! s the notaton for dvded powers. The Kashwara operator ẽ assocated wth E s defned analogously. By [Kas91] there exsts a crystal bass (L(λ), B(λ)) for the smple U q (sl n (C))-module L q (λ). Here L(λ) denotes the crystal lattce, the mnmal lattce over the ratonal functons regular at 0 that contans a hghest weght vector v λ of L q (λ) and that s stable under the acton of the Kashwara operators f, ẽ. The subset B(λ) of L(λ)/qL(λ) s gven by all nonzero elements of the form f 1... f r (v λ ) + ql(λ). 31

32 I.1. Crystal bases and partcle confguratons One defnes the crystal graph to be an orented graph wth vertces B(λ) and edges labelled by 1,..., n 1, so that there s an -labelled edge from b to b B(λ) f and only f f (b) = b modulo ql(λ). Ths s the case f and only f ẽ (b ) = b modulo ql(λ). By abuse of notaton, the crystal graph s also denoted by B(λ). Crystal bases are partcularly sutable for the computaton of tensor products. Gven L q (λ) wth crystal bass (L(λ), B(λ)) and L q (λ ) wth crystal bass (L(λ ), B(λ )), one can easly determne a crystal graph for L(λ) C(q) L(λ ) on the set of vertces B(λ) B(λ ) = B(λ) B(λ ). The tensor product rule prescrbes on whch tensor factor the Kashwara operator f acts, see [HK02, Theorem 4.4.1]. In type A n, for any smple U q (sl n (C))-module L q (λ), the set B(λ) can be realzed by semstandard Young tableaux of shape λ wth entres 1,..., n. The hghest weght vector v λ of L q (λ) s represented by the standard semstandard Young tableau of shape λ where all entres n the kth row are equal to k. In the crystal graph B(λ), f two semstandard Young tableaux are connected by an -labelled edge, then ther entres are the same except that n one box the entry s replaced by +1. Let us recall the detals: In Fgure I we depct the crystal graph for the standard/vector representaton L q (ω 1 ) of U q (sl n (C)). As a C(q)-vector space, L q (ω 1 ) C(q) n n 1 n Fgure I : The crystal graph for the standard/vector representaton of U q (sl n (C)). Here, a crystal bass of L q (ω 1 ) s gven by (L(ω 1 ), B(ω 1 )). The crystal lattce L(ω 1 ) s spanned over the ratonal functons regular at 0 by the standard bass vectors v 1,..., v n on whch F acts by F v = v +1, F v j = 0 for j. The set B(ω 1 ) s gven by the resdue classes of the standard bass vectors n L(ω 1 )/ql(ω 1 ). In Fgure I a box wth entry s dentfed wth the resdue class of the th standard bass vector v of C(q) n. The vertces of the crystal graph of L q (λ) can be dentfed wth the set of semstandard Young tableaux of shape λ as follows: The tensor product rule allows to compute the crystal graph of L q (ω 1 ) (λ λ n 1 ). Then the crystal graph B(λ) s dentfed wth a connected component n B(ω 1 ) (λ λ n 1 ) by an admssble readng. The tensor product rule prescrbes on whch tensor factor the Kashwara operator f acts, hence, n whch box the entry s turned nto + 1. In general there are many possble choces of admssble readngs, but the crystal structure on B(λ) does not depend on ths choce, see [HK02, Theorem 7.3.6]. 32

33 I.1.1. Quantum groups and crystal bases of type A n and Ân For a tensor product a 1 a 2... a d n B(ω1 ) d, the tensor product rule can be summarzed as follows: We need to determne the box wth entry on whch f has to act. The boxes wth entres j, + 1 are rrelevant and thus removed. Then all ncreasng par of boxes are removed, that s, a box wth entry a r = whch s followed mmedately by a box wth entry a r+s = + 1 n the remanng tensor product (where all boxes wth entres a r+1,..., a r+s 1 have been prevously removed). Ths process s repeated for the remanng tensor factors untl no ncreasng par of boxes remans. If the fnal result does not contan any box wth entry, then f acts by zero. If there are some boxes wth entry left, then f acts on the leftmost such box Fgure I : Examples of crystal graphs for L q (3ε 1 ), L q (2ε 1 + ε 2 ), L q (ε 1 + ε 2 ) U q (sl 3 (C)) mod and L q (ε 1 + ε 2 ) U q (sl 5 (C)) mod. 33

34 I.1. Crystal bases and partcle confguratons The examples n Fgure I llustrate the crystal graphs for the U q (sl 3 (C))-modules L q (3ε 1 ), L q (2ε 1 + ε 2 ), L q (ε 1 + ε 2 ), and for the U q (sl 5 (C))-module L q (ε 1 + ε 2 ). L q (2ε 1 + ε 2 ) ths s Example from [HK02]. The crystal graphs B(λ) for λ = kω 1 = and λ = ω k = are specal. Recall that B(kω 1 ) s the crystal graph correspondng to the symmetrc representaton Sym k (C n ) of sl n (C), and B(ω k ) s the crystal graph for the alternatng representaton Λ k (C n ). The tensor product rule s partcularly easy to formulate for B(kω 1 ) and B(ω k ), and the acton of the Kashwara operators on semstandard Young tableaux of shape kω 1 or ω k s ndependent of the relatve postons of the boxes wth entres, + 1. Let us dscuss ths n detal: Let us start wth Young tableaux of shape kω 1. There s precsely one admssble readng. It s gven by k k For Snce the sequence 1, 2,..., k {1,..., n} s (weakly) ncreasng, the reversed sequence obtaned from the admssble readng k,..., 2, 1 s decreasng. In partcular, there are no ncreasng pars of boxes. In ths case, the tensor product rule for crystals smply amounts to the followng rule: I Lemma. Let 1 n 1 and k Z 0. On semstandard Young tableaux of shape kω 1 that contan a box wth entry the Kashwara operator f acts on the rghtmost box wth entry, replacng t by + 1. On semstandard Young tableaux of shape kω 1 that do not contan any box wth entry the Kashwara operator f acts by zero. For Young tableaux of shape ω k, there s precsely one admssble readng gven by k k. The sequence 1, 2,..., k {1,..., n} s strctly ncreasng. In partcular, no entry s repeated, and a quck case-by-case analyss gves the followng rule equvalent to the tensor product rule: I Lemma. Let 1 n 1 and 1 k n 1. On semstandard Young tableaux of shape ω k that contan a box wth entry and that do not contan any box wth entry + 1 the Kashwara operator f acts on (the only) box wth entry, replacng t by

35 I.1.1. Quantum groups and crystal bases of type A n and Ân On any other semstandard Young tableaux of shape ω k the Kashwara operator f acts by zero. I Remark. The rules from Lemma I and Lemma I are formulated ndependently of the relatve postons of the boxes wth entres, + 1. I Remark. For hooks of the form there s only one admssble readng, too. But t s not guaranteed that the sequence we obtan from the admssble readng s decreasng or strctly ncreasng, and the result of the applcaton of f depends on the exact postons of the boxes wth entres, + 1 n the Young tableau. For example, n Fgure I we have seen that f 2 maps to 1 3 3, whereas s mapped to zero. Let us now nvestgate some of the relatons among the Kashwara operators f. I Lemma. ) Let k Z 0. On B(kω 1 ) {0} we have f fj = f j f for all 1, j n 1 so that j > 1, f f 1 f = f 2 f 1 for all 2 n 1, f f+1 f = f 2 +1 f for all 1 n 2, f f 1 f+1 f = f 2 +1 f f 1 for all 2 n 2. ) Let 1 k n 1. On B(ω k ) {0} we have f fj = f j f for all 1, j n 1 so that j > 1, f 2 = 0 for all 1 n 1, f f 1 f = 0 for all 2 n 1, f f+1 f = 0 for all 1 n 2. Proof. Ths follows from the explct realsaton of the Kashwara operators n Lemma I and Lemma I In partcular, Lemma I mples that the relatons f f 1 f = f 2 f 1 and f f+1 f = f +1 f 2 (I.1.1) hold for all crystals B(kω 1 ) and B(ω k ). In contrast, the relaton f f 1 f+1 f = f 2 +1 f f 1 s specal for B(kω 1 ) and does not hold for B(ω k ). For example, for n = 5 and ω 2 we have f 2 f1 f3 f2 ( 1 2 ) =