673 Recent Developments in Representation Theory Maurice Auslander Distinguished Lectures and International Conference May 1–6, 2014 Woods Hole Oceanographic Institute, Woods Hole, MA Alex Martsinkovsky Gordana Todorov Kiyoshi Igusa Editors AmericanMathematicalSociety 673 Recent Developments in Representation Theory Maurice Auslander Distinguished Lectures and International Conference May 1–6, 2014 Woods Hole Oceanographic Institute, Woods Hole, MA Alex Martsinkovsky Gordana Todorov Kiyoshi Igusa Editors AmericanMathematicalSociety Providence,RhodeIsland EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Catherine Yan 2010 Mathematics Subject Classification. Primary 16G10,16G20, 16G70, 16T05, 14N15. Library of Congress Cataloging-in-Publication Data Names: Maurice Auslander Distinguished Lectures and International Conference (2014: Woods Hole, Mass.) Martsinkovsky, A. (Alex), editor. — Todorov, G. (Gordana), editor. — Igusa, Kiyoshi,1949-editor. Title: Recentdevelopmentsinrepresentationtheory: MauriceAuslanderDistinguishedLectures andInternationalConference: May1-6,2014,WoodsHoleOceanographicInstitute,WoodsHole, Massachusetts/AlexMartsinkovsky,GordanaTodorov,KiyoshiIgusa,editors. Description: Providence, Rhode Island: AmericanMathematicalSociety, [2016]—Series: Con- temporarymathematics;volume673—Includesbibliographicalreferences. Identifiers: LCCN2016001389—ISBN9781470419554(alk. paper) Subjects: LCSH:Associativerings–Congresses. —Representationsofrings(Algebra)–Congresses. — AMS: Associative rings and algebras – Representation theory of rings and algebras – Repre- sentations of Artinian rings. msc — Associative rings and algebras – Representation theory of rings and algebras – Representations of quivers and partially ordered sets. msc — Associative rings and algebras – Representation theory of rings and algebras – Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers. msc — Associative rings and algebras – Hopf algebras, quantum groups and related topics – Hopf algebras and their applications. msc — Algebraic geometry – Projective and enumerative geometry – Classical problems, Schubert calculus. msc Classification: LCCQA251.5.M282014—DDC515/.7223–dc23 LCrecordavailableathttp://lccn.loc.gov/2016001389 ContemporaryMathematicsISSN:0271-4132(print);ISSN:1098-3627(online) DOI:http://dx.doi.org/10.1090/conm/673 Copying and reprinting. Individual readersofthispublication,andnonprofit librariesacting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews,providedthecustomaryacknowledgmentofthesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink(cid:2) service. Formoreinformation,pleasevisit: http://www.ams.org/rightslink. Sendrequestsfortranslationrightsandlicensedreprintstoreprint-permission@ams.org. Excludedfromtheseprovisionsismaterialforwhichtheauthorholdscopyright. Insuchcases, requestsforpermissiontoreuseorreprintmaterialshouldbeaddresseddirectlytotheauthor(s). Copyrightownershipisindicatedonthecopyrightpage,oronthelowerright-handcornerofthe firstpageofeacharticlewithinproceedingsvolumes. (cid:2)c 2016bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 212019181716 Contents Preface v Orders in Artinian rings, Goldie’s Theorem and the largest left quotient ring of a ring V. V. Bavula 1 Invariant theory of Artin-Schelter regular algebras: a survey Ellen E. Kirkman 25 The Catalan combinatorics of the hereditary artin algebras Claus Michael Ringel 51 Grassmannians, flag varieties, and Gelfand-Zetlin polytopes Evgeny Smirnov 179 On the combinatorics of the set of tilting modules Luise Unger 227 iii Preface This volume continues thenewly establishedtraditionofpublishing expository lecturesfromtheMauriceAuslanderDistinguishedLecturesandInternationalCon- ference. These events took place May 1-6, 2014 at the Woods Hole Oceanographic Institute in Falmouth, MA. The support from the National Science Foundation (Grant DMS-1162304) al- lowedustoinvite59participantsfromAustria,Canada,CzechRepublic,Germany, India, Mexico, Norway, Russia, Spain, Turkey, the UK, and the US. Twenty five of them were either graduate students or postdocs. The talks in the conference spread over a wide area of research. The same appliestotheselectedexpositorylecturesinthisvolume. VladimirBavulapresents a survey of old and new results on left orders in left Artinian rings. Ellen Kirkman gives an introduction to invariant theory of Artin-Schelter regular algebras. Claus- MichaelRingeldescribesthelatestresultsrelatingcombinatoricsandrepresentation theory of hereditary artin algebras. Evgeny Smirnov's survey touches upon the classical topic of Schubert calculus on Grassmannian and flag varieties and relates it to the combinatorics of Gelfand-Zetlin polytopes. Luise Unger introduces the reader to the combinatorics of tilting modules. The list of the topics above, as spread out as it is, does not fully represent the scientific breadth of the conference, even in the expository aspect of it. Because of the time constraints, some of the presented expository lectures did not transition into a written format. The editors hope very much that at least some of them will eventually make it into print. The support for the Conference from the NSF and the multi-year continued support for the Maurice Auslander Distinguished Lecturesfrom BerniceAuslander create a unique symbiotic relationship between the two events. The expository component of the Conference has been attracting a large number of graduate stu- dents and young researchers. The editors of this volume hope that its publication will benefit many more mathematicians around the world. The Editors v ContemporaryMathematics Volume673,2016 http://dx.doi.org/10.1090/conm/673/13488 Orders in Artinian rings, Goldie’s Theorem and the largest left quotient ring of a ring V. V. Bavula Abstract. Thisshortsurveyisaboutsomeoldandnewresultsonleftorders inleftArtinianrings,newcriteriaforaringtohaveasemisimpleleftquotient ring,newconcepts(e.g.,thelargestleftquotientringofaring). Contents (1) Introduction. (2) New criteria for a ring to have a semisimple left quotient ring. (3) Old criteria for a ring to have a left Artinian left quotient ring. (4) Necessary and sufficient conditions for a ring to have a left Artinian left quotient ring. (5) A criterion via associated graded ring. (6) Criteria similar to Robson’s Criterion. (7) A left quotient ring of a factor ring. (8) The largest denominator sets and the largest left quotient ring of a ring. (9) The maximal left quotient rings of a ring. (10) Examples. 1. Introduction Inthispaper,modulemeansaleftmodule,allringsareassociativewith1. The present paper comprises three parts. Part I, ‘New Criteria for a Ring to have a Semisimple Left Quotient Ring’ (Section 2). Goldie’s Theorem (1958, 1960) is an old and up to 2013 was the only example of such criteria. Four new criteria will be given that are independent of Goldie’s Theorem and are based on completely new ideas and approach. Part II, ‘Left Orders in Left Artinian Rings’ (Sections 3–7), deals with old and new criteria for a ring to have a left Artinian left quotient ring. 2010 Mathematics Subject Classification. Primary 16U20, 16P40, 16S32, 13N10, 16P20, 16U20,16P60. Key words and phrases. Goldie’s Theorem, orders, left Artinian ring, the left quotient ring ofaring,thelargestleftquotientringofaring,thelargestregularleftOreset. (cid:2)c2016 American Mathematical Society 1 2 V.V.BAVULA Part III, ‘The Largest Left Quotient Ring of a Ring’ (Sections 8–10), is about new recent concepts and results obtained in order to answer the old question: Why does the classical left quotient ring of a ring not always exist? A positive step in this direction is the fact that for an arbitrary ring R there alwaysexiststhelargestleftquotientringQ (R),[4],whichcoincideswiththeclas- l sicalleftquotientringifthelatterexists. Anothernewconcept/factistheexistence of the maximal left quotient rings (for an arbitrary ring R). Their existence gives an affirmative answer to the following question: given a ring R, replace the ring R byitsleftlocalizationS−1Rataleftdenominatorset(thatnotnecessarilyconsists 1 of regular elements). Then repeat the step again and again (infinitely many times of arbitrary cardinality, if necessary): S−1(S−1R), S−1(S−1(S−1R)),...; will this 2 1 3 2 1 process stop? (i.e. do we reach the moment we cannot invert anything new?) The answer is yes and the rings we obtain are called the maximal left quotient rings of a ring and any such a ring can be written as S−1R for some left denominator set S of the ring R, [4]. Goldie’s Theorem (1960), which is one of the most important results in Ring Theory, is a criterion for a ring to have a semisimple left quotient ring. The aim of thepaperistogivefournewcriteria(usingacompletelydifferentapproachandnew ideas). The first one is based on the recently discovered fact that for an arbitrary ring R the set M of maximal left denominator sets of R is a non-empty set [4]. Theorem (The Firs(cid:2)t Criterion). A ring R has a semisimple left quotient ring Q iff M is a finite set, ass(S) = 0 and, for each S ∈ M, the ring S−1R is S∈M (cid:3) a simple left Artinian ring. In this case, Q(cid:4) S−1R. S∈M The Second Criterion is given via the minimal primes of R and goes further than the First one in the sense that it describes explicitly the maximal left denom- inator sets S via the minimal primes of R. The Third Criterion is close to Goldie’s Criterionbutitiseasiertocheckinapplications(basically,itreducesGoldie’sThe- oremtotheprimecase). TheFourthCriterionisgivenviacertainleftdenominator sets. The conditions in old criteria for a ring R to have a left Artinian left quotient ring Q are ‘strong’ (like ‘the ring R is a left Goldie ring’) and given in terms of the ring R itself and its ideals. The conditions of the new criteria are ‘weak’ (like ‘the ring R is a left Goldie ring’ where R := R/n and n is the prime radical of R) and given in terms of the ring R (rather than R) and of its finitely many explicit modules. Goldie’s Theorem [12] characterizes left orders in semi-simple rings, it is a criterion of when the left quotient ring of a ring is semi-simple (earlier, character- izations were given, by Goldie [11] and Lesieur and Croisot [16], of left orders in a simple Artinian ring). Talintyre [26] and Feller and Swokowski [10] have given conditionswhicharesufficientforaleftNoetherianringtohavealeftquotientring. Further, foraleftNoetherianringwhichhasaleftquotientring, Talintyre[27]has established necessary and sufficient conditions for the left quotient ring to be left Artinian. Small [21,22], Robson [20], and latter Tachikawa [25] and Hajarnavis [13]havegivendifferentcriteriaforaringtohavealeftArtinianleftquotientring. In this paper, three more new criteria are given (Theorem 4.1, Theorem 5.1 and Theorem 6.1). ORDERS IN ARTINIAN RINGS 3 Theorem 7.1 gives an affirmative answer to the question: Let R be a ring with a left Artinian left quotient ring Q and I be a C-closed ideal of R such that I ⊆n. Is the left quotient ring Q(R/I) of R/I a left Artinian ring? The set C of regular elements of a ring R is not always a left (or right) Ore set in R (hence, the left quotient ring C−1R or the right quotient ring RC−1 does not always exist) but there always exists the largest regular left Ore set S and l,0 the largest regular right Ore set S in C of the ring R, [4]. In general, S (cid:6)=S , r,0 l,0 r,0 [3]. In [4], the largest left quotient ring Q (R) := S−1R and the largest right l l,0 quotient ring Qr(R(cid:4)) := RSr−,01 are introduced. In [3], these rings are found for the ring I = K(cid:7)x,∂, (cid:8) of polynomial integro-differential operators over a field K of 1 characteristic zero, S (I )(cid:6)=S (I ) and Q (I )(cid:6)(cid:4)Q (I ). l,0 1 r,0 1 l 1 r 1 Part 1. New criteria for a ring to have a semisimple left quotient ring In the paper, the following notation is fixed: • Risaringwith1, R∗ isitsgroupofunits, n=n isitsprimeradicaland R Min(R) is the set of minimal primes of R; • C = C is the set of regular elements of the ring R (i.e. C is the set of R non-zero-divisors of the ring R); • Q = Q (R) := C−1R is the left quotient ring (the classical left ring of l,cl fractions) of the ring R (if it exists) and Q∗ is the group of units of Q; • Den (R,a)isthesetofleftdenominatorsetsS ofRwithass(S)=awhere l a is an ideal of R and ass(S):={r ∈R|sr =0 for some s∈S}; • max.Den (R)isthesetofmaximalleftdenominatorsetsofR(itisalways l a non-empty set, [4]); • Ore (R):={S|S is a left Ore set in R}; l • Den (R):={S|S is a left denominator set in R}; l • Loc (R):={S−1R|S ∈Den (R)}; l l • Ass (R):={ass(S)|S ∈Den(R)}; l l • S = S (R) = S (R) is the largest element of the poset (Den(R,a),⊆) a a l,a l and Q (R):=Q (R):=S−1R is the largest left quotient ring associated a l,a a to a, S exists (Theorem 2.1, [4]); a • In particular, S = S (R) = S (R) is the largest element of the poset 0 0 l,0 (Den(R,0),⊆), i.e. the largest regular left Ore set of R, and Q (R) := l l S−1R is the largest left quotient ring of R [4]; 0 • Loc (R) := {[S−1R]|S ∈ Den (R)} where [S−1R] is an R-isomorphism l l class of the ring S−1R (a ring isomorphism σ : S−1R → S(cid:5)−1R is called an R-isomorphism if σ(r) = r for all elements r ∈ R); we usually write 1 1 S−1R instead of [S−1R] if this does not lead to confusion; • Loc (R,a):={[S−1R]|S ∈Den (R,a)}. l l The largestleftquotientringof aring. LetRbearing. Amultiplicatively closed subset S of R (i.e. a multiplicative sub-semigroup of (R,·) such that 1 ∈ S and 0(cid:6)∈S) is said to be a left Ore set if it satisfies the left Ore condition: for each r ∈R and s∈S, (cid:5) Sr Rs(cid:6)=∅.