Andrei Zelevinsky, 1953–2013
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COMBINATORIAL Bn-ANALOGUES of SCHUBERT POLYNOMIALS 0
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 348, Number 9, September 1996 COMBINATORIAL Bn-ANALOGUES OF SCHUBERT POLYNOMIALS SERGEY FOMIN AND ANATOL N. KIRILLOV Abstract. Combinatorial Bn-analogues of Schubert polynomials and corre- sponding symmetric functions are constructed and studied. The development is based on an exponential solution of the type B Yang-Baxter equation that involves the nilCoxeter algebra of the hyperoctahedral group. 0. Introduction This paper is devoted to the problem of constructing type B analogues of the Schubert polynomials of Lascoux and Sch¨utzenberger (see, e.g., [L2], [M] and the literature therein). We begin with reviewing the basic properties of the type A polynomials, stating them in a form that would allow subsequent generalizations to other types. Let W = Wn be the Coxeter group of type An with generators s1, ... ,sn (that is, the symmetric group Sn+1). Let x1, x2, ... be formal variables. Then W naturally acts on the polynomial ring C[x1,...,xn+1] by permuting the variables. Let IW denote the ideal generated by homogeneous non-constant W -symmetric polynomials. By a classical result [Bo], the cohomology ring H(F ) of the flag variety of type A can be canonically identified with the quotient C[x1,...,xn+1]/IW .This ring is graded by the degree and has a distinguished linear basis of homogeneous cosets Xw modulo IW , labelled by the elements w of the group. Let us state for the record that, for an element w W of length l(w), ∈ (0) Xw is a homogeneous polynomial of degree l(w); X1=1; the latter condition signifies proper normalization. -
Algebra & Number Theory Vol. 7 (2013)
Algebra & Number Theory Volume 7 2013 No. 3 msp Algebra & Number Theory msp.org/ant EDITORS MANAGING EDITOR EDITORIAL BOARD CHAIR Bjorn Poonen David Eisenbud Massachusetts Institute of Technology University of California Cambridge, USA Berkeley, USA BOARD OF EDITORS Georgia Benkart University of Wisconsin, Madison, USA Susan Montgomery University of Southern California, USA Dave Benson University of Aberdeen, Scotland Shigefumi Mori RIMS, Kyoto University, Japan Richard E. Borcherds University of California, Berkeley, USA Raman Parimala Emory University, USA John H. Coates University of Cambridge, UK Jonathan Pila University of Oxford, UK J-L. Colliot-Thélène CNRS, Université Paris-Sud, France Victor Reiner University of Minnesota, USA Brian D. Conrad University of Michigan, USA Karl Rubin University of California, Irvine, USA Hélène Esnault Freie Universität Berlin, Germany Peter Sarnak Princeton University, USA Hubert Flenner Ruhr-Universität, Germany Joseph H. Silverman Brown University, USA Edward Frenkel University of California, Berkeley, USA Michael Singer North Carolina State University, USA Andrew Granville Université de Montréal, Canada Vasudevan Srinivas Tata Inst. of Fund. Research, India Joseph Gubeladze San Francisco State University, USA J. Toby Stafford University of Michigan, USA Ehud Hrushovski Hebrew University, Israel Bernd Sturmfels University of California, Berkeley, USA Craig Huneke University of Virginia, USA Richard Taylor Harvard University, USA Mikhail Kapranov Yale University, USA Ravi Vakil Stanford University, -
2010 Integral
Autumn 2010 Volume 5 Massachusetts Institute of Technology 1ntegral n e w s f r o m t h e mathematics d e p a r t m e n t a t m i t The retirement of seven of our illustrious col- leagues this year—Mike Artin, David Ben- Inside ney, Dan Kleitman, Arthur Mattuck, Is Sing- er, Dan Stroock, and Alar Toomre—marks • Faculty news 2–3 a shift to a new generation of faculty, from • Women in math 3 those who entered the field in the Sputnik era to those who never knew life without email • A minority perspective 4 and the Internet. The older generation built • Student news 4–5 the department into the academic power- house it is today—indeed they were the core • Funds for RSI and SPUR 5 of the department, its leadership and most • Retiring faculty and staff 6–7 distinguished members, during my early years at MIT. Now, as they are in the process • Alumni corner 8 of retiring, I look around and see that my con- temporaries are becoming the department’s Dear Friends, older group. Yikes! another year gone by and what a year it Other big changes are in the works. Two of Marina Chen have taken the lead in raising an was. We’re getting older, and younger, cele- our dedicated long-term administrators— endowment for them. Together with those of brating prizes and long careers, remembering Joanne Jonsson and Linda Okun—have Tim Lu ’79 and Peiti Tung ’79, their commit- our past and looking to the future. -
Pure Mathematics
Proceedings of Symposia in PURE MATHEMATICS Volume 101 Representations of Reductive Groups Conference in honor of Joseph Bernstein Representation Theory & Algebraic Geometry June 11–16, 2017 Weizmann Institute of Science, Rehovot, Israel and The Hebrew University of Jerusalem, Is- rael Avraham Aizenbud Dmitry Gourevitch David Kazhdan Erez M. Lapid Editors 10.1090/pspum/101 Volume 101 Representations of Reductive Groups Conference in honor of Joseph Bernstein Representation Theory & Algebraic Geometry June 11–16, 2017 Weizmann Institute of Science, Rehovot, Israel and The Hebrew University of Jerusalem, Is- rael Avraham Aizenbud Dmitry Gourevitch David Kazhdan Erez M. Lapid Editors Proceedings of Symposia in PURE MATHEMATICS Volume 101 Representations of Reductive Groups Conference in honor of Joseph Bernstein Representation Theory & Algebraic Geometry June 11–16, 2017 Weizmann Institute of Science, Rehovot, Israel and The Hebrew University of Jerusalem, Is- rael Avraham Aizenbud Dmitry Gourevitch David Kazhdan Erez M. Lapid Editors 2010 Mathematics Subject Classification. Primary 11F27, 11F70, 20C08, 20C33, 20G05, 20G20, 20G25, 22E47. Library of Congress Cataloging-in-Publication Data Names: Conference on Representation Theory and Algebraic Geometry in honor of Joseph Bern- stein (2017 : Jerusalem, Israel) | Aizenbud, Avraham, 1983- editor. | Gourevitch, Dmitry, 1981- editor. | Kazhdan, D. A. (David A.), 1946- editor. | Lapid, Erez, 1971- editor. | Bernstein, Joseph, 1945- honoree. Title: Representations of reductive groups : Conference on Representation Theory and Algebraic Geometry in honor of Joseph Bernstein, June 11-16, 2017, The Weizmann Institute of Sci- ence & The Hebrew University of Jerusalem, Jerusalem, Israel / Avraham Aizenbud, Dmitry Gourevitch, David Kazhdan, Erez M. Lapid, editors. Description: Providence, Rhode Island : American Mathematical Society, [2019] | Series: Pro- ceedings of symposia in pure mathematics ; Volume 101 | Includes bibliographical references. -
Scientific Report for the Year 2000
The Erwin Schr¨odinger International Boltzmanngasse 9 ESI Institute for Mathematical Physics A-1090 Wien, Austria Scientific Report for the Year 2000 Vienna, ESI-Report 2000 March 1, 2001 Supported by Federal Ministry of Education, Science, and Culture, Austria ESI–Report 2000 ERWIN SCHRODINGER¨ INTERNATIONAL INSTITUTE OF MATHEMATICAL PHYSICS, SCIENTIFIC REPORT FOR THE YEAR 2000 ESI, Boltzmanngasse 9, A-1090 Wien, Austria March 1, 2001 Honorary President: Walter Thirring, Tel. +43-1-4277-51516. President: Jakob Yngvason: +43-1-4277-51506. [email protected] Director: Peter W. Michor: +43-1-3172047-16. [email protected] Director: Klaus Schmidt: +43-1-3172047-14. [email protected] Administration: Ulrike Fischer, Eva Kissler, Ursula Sagmeister: +43-1-3172047-12, [email protected] Computer group: Andreas Cap, Gerald Teschl, Hermann Schichl. International Scientific Advisory board: Jean-Pierre Bourguignon (IHES), Giovanni Gallavotti (Roma), Krzysztof Gawedzki (IHES), Vaughan F.R. Jones (Berkeley), Viktor Kac (MIT), Elliott Lieb (Princeton), Harald Grosse (Vienna), Harald Niederreiter (Vienna), ESI preprints are available via ‘anonymous ftp’ or ‘gopher’: FTP.ESI.AC.AT and via the URL: http://www.esi.ac.at. Table of contents General remarks . 2 Winter School in Geometry and Physics . 2 Wolfgang Pauli und die Physik des 20. Jahrhunderts . 3 Summer Session Seminar Sophus Lie . 3 PROGRAMS IN 2000 . 4 Duality, String Theory, and M-theory . 4 Confinement . 5 Representation theory . 7 Algebraic Groups, Invariant Theory, and Applications . 7 Quantum Measurement and Information . 9 CONTINUATION OF PROGRAMS FROM 1999 and earlier . 10 List of Preprints in 2000 . 13 List of seminars and colloquia outside of conferences . -
Volume 8 Number 4 October 1995
VOLUME 8 NUMBER 4 OCTOBER 1995 AMERICANMATHEMATICALSOCIETY EDITORS William Fulton Benedict H. Gross Andrew Odlyzko Elias M. Stein Clifford Taubes ASSOCIATE EDITORS James G. Arthur Alexander Beilinson Louis Caffarelli Persi Diaconis Michael H. Freedman Joe Harris Dusa McDuff Hugh L. Montgomery Marina Ratner Richard Schoen Richard Stanley Gang Tian W. Hugh Woodin PROVIDENCE, RHODE ISLAND USA ISSN 0894-0347 Journal of the American Mathematical Society This journal is devoted to research articles of the highest quality in all areas of pure and applied mathematics. Subscription information. The Journal of the American Mathematical Society is pub- lished quarterly. Subscription prices for Volume 8 (1995) are $158 list, $126 institutional member, $95 individual member. Subscribers outside the United States and India must pay a postage surcharge of $8; subscribers in India must pay a postage surcharge of $18. Expedited delivery to destinations in North America $13; elsewhere $36. Back number information. For back issues see the AMS Catalog of Publications. Subscriptions and orders should be addressed to the American Mathematical Society, P. O. Box 5904, Boston, MA 02206-5904. All orders must be accompanied by payment. Other correspondence should be addressed to P. O. Box 6248, Providence, RI 02940- 6248. Copying and reprinting. Material in this journal may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. -
Arxiv:Math/9912128V1 [Math.RA] 15 Dec 1999 Iigdiagrams
TOTAL POSITIVITY: TESTS AND PARAMETRIZATIONS SERGEY FOMIN AND ANDREI ZELEVINSKY Introduction A matrix is totally positive (resp. totally nonnegative) if all its minors are pos- itive (resp. nonnegative) real numbers. The first systematic study of these classes of matrices was undertaken in the 1930s by F. R. Gantmacher and M. G. Krein [20, 21, 22], who established their remarkable spectral properties (in particular, an n × n totally positive matrix x has n distinct positive eigenvalues). Earlier, I. J. Schoenberg [41] discovered the connection between total nonnegativity and the following variation-diminishing property: the number of sign changes in a vec- tor does not increase upon multiplying by x. Total positivity found numerous applications and was studied from many dif- ferent angles. An incomplete list includes: oscillations in mechanical systems (the original motivation in [22]), stochastic processes and approximation theory [25, 28], P´olya frequency sequences [28, 40], representation theory of the infinite symmetric group and the Edrei-Thoma theorem [13, 44], planar resistor networks [11], uni- modality and log-concavity [42], and theory of immanants [43]. Further references can be found in S. Karlin’s book [28] and in the surveys [2, 5, 38]. In this paper, we focus on the following two problems: (i) parametrizing all totally nonnegative matrices; (ii) testing a matrix for total positivity. Our interest in these problems stemmed from a surprising representation-theoretic connection between total positivity and canonical bases for quantum groups, dis- covered by G. Lusztig [33] (cf. also the surveys [31, 34]). Among other things, he extended the subject by defining totally positive and totally nonnegative elements for any reductive group. -
SCIENTIFIC REPORT for the 5 YEAR PERIOD 1993–1997 INCLUDING the PREHISTORY 1991–1992 ESI, Boltzmanngasse 9, A-1090 Wien, Austria
The Erwin Schr¨odinger International Boltzmanngasse 9 ESI Institute for Mathematical Physics A-1090 Wien, Austria Scientific Report for the 5 Year Period 1993–1997 Including the Prehistory 1991–1992 Vienna, ESI-Report 1993-1997 March 5, 1998 Supported by Federal Ministry of Science and Transport, Austria http://www.esi.ac.at/ ESI–Report 1993-1997 ERWIN SCHRODINGER¨ INTERNATIONAL INSTITUTE OF MATHEMATICAL PHYSICS, SCIENTIFIC REPORT FOR THE 5 YEAR PERIOD 1993–1997 INCLUDING THE PREHISTORY 1991–1992 ESI, Boltzmanngasse 9, A-1090 Wien, Austria March 5, 1998 Table of contents THE YEAR 1991 (Paleolithicum) . 3 Report on the Workshop: Interfaces between Mathematics and Physics, 1991 . 3 THE YEAR 1992 (Neolithicum) . 9 Conference on Interfaces between Mathematics and Physics . 9 Conference ‘75 years of Radon transform’ . 9 THE YEAR 1993 (Start of history of ESI) . 11 Erwin Schr¨odinger Institute opened . 11 The Erwin Schr¨odinger Institute An Austrian Initiative for East-West-Collaboration . 11 ACTIVITIES IN 1993 . 13 Short overview . 13 Two dimensional quantum field theory . 13 Schr¨odinger Operators . 16 Differential geometry . 18 Visitors outside of specific activities . 20 THE YEAR 1994 . 21 General remarks . 21 FTP-server for POSTSCRIPT-files of ESI-preprints available . 22 Winter School in Geometry and Physics . 22 ACTIVITIES IN 1994 . 22 International Symposium in Honour of Boltzmann’s 150th Birthday . 22 Ergodicity in non-commutative algebras . 23 Mathematical relativity . 23 Quaternionic and hyper K¨ahler manifolds, . 25 Spinors, twistors and conformal invariants . 27 Gibbsian random fields . 28 CONTINUATION OF 1993 PROGRAMS . 29 Two-dimensional quantum field theory . 29 Differential Geometry . 29 Schr¨odinger Operators . -
The Yang-Baxter Equation, Symmetric Functions, and Schubert Polynomials
DISCRETE MATHEMATICS ELSEVIER Discrete Mathematics 153 (1996) 123 143 The Yang-Baxter equation, symmetric functions, and Schubert polynomials Sergey Fomin a'b'*, Anatol N. Kirillov c a Department of Mathematics, Room 2-363B, Massachusetts Institute of Technolooy, Cambridge, MA 02139, USA b Theory of Aloorithms Laboratory, SPIIRAN, Russia c Steklov Mathematical Institute, St. Petersbur9, Russia Received 31 August 1993; revised 18 February 1995 Abstract We present an approach to the theory of Schubert polynomials, corresponding symmetric functions, and their generalizations that is based on exponential solutions of the Yang-Baxter equation. In the case of the solution related to the nilCoxeter algebra of the symmetric group, we recover the Schubert polynomials of Lascoux and Schiitzenberger, and provide simplified proofs of their basic properties, along with various generalizations thereof. Our techniques make use of an explicit combinatorial interpretation of these polynomials in terms of configurations of labelled pseudo-lines. Keywords: Yang-Baxter equation; Schubert polynomials; Symmetric functions 1. Introduction The Yang-Baxter operators hi(x) satisfy the following relations (cf. [1,7]): hi(x)hj(y) = hj(y)hi(x) if li -j[ >_-2; hi(x)hi+l (x + y)hi(y) = hi+l (y)hi(x + y)hi+l(X); The role the representations of the Yang-Baxter algebra play in the theory of quantum groups [9], the theory of exactly solvable models in statistical mechanics [1], low- dimensional topology [7,27,16], the theory of special functions, and other branches of mathematics (see, e.g., the survey [5]) is well known. * Corresponding author. 0012-365X/96/$15.00 (~) 1996--Elsevier Science B.V. -
From Algebras to Varieties
F EATURE Principal Investigator Alexey Bondal Research Area:Mathematics From Algebras to Varieties geometry of varieties superlatively developed by Derived categories after Grothendieck Italian school of the late 19th and early 20th century. Both groups had very strong representatives, but Homological algebra is considered by they had scarce overlap in research. mathematicians to be one of the most formal Perhaps, one of the reasons for this strange state subjects within mathematics. Its formal austerity of affairs was a side effect of the great achievement requires a lot of efforts to come through basic of one of the best Grothendieck’s students, Pierre denitions and frightens off many of those who Deligne, who used complicated homological algebra starts studying the subject. This high level of to prove Weil conjectures. These conjectures are, formality of the theory gives an impression that it probably, more of arithmetic nature and do not is a ding an sich, something that has no possible have so much to do with geometry of algebraic way of comprehension for outsiders and efforts for varieties in the classical sense of Italian school. learning the theory would never be paid back. For years, applications of derived categories were After Alexander Grothendieck, the great creator developed rather in the area of number theoretical of modern homological algebra who introduced the and topological aspects of varieties and, later, in concept of derived categories, had left the stage Representation Theory than in classical geometry of where he occupied a central place for decades, the varieties. opinion that the homological theory had reached its bounds and become a useless formal theory Associative algebras was widely spread in the mathematical community. -
Arxiv:2104.14530V2 [Math.RT]
REFLECTION LENGTH WITH TWO PARAMETERS IN THE ASYMPTOTIC REPRESENTATION THEORY OF TYPE B/C AND APPLICATIONS MAREK BOZEJKO,˙ MACIEJ DOŁ ˛EGA, WIKTOR EJSMONT, AND SWIATOSŁAW´ R. GAL ABSTRACT. We introduce a two-parameter function φq+,q− on the infinite hyperoctahedral group, which is a bivariate refinement of the reflection length. We show that this signed reflection function φq+,q− is positive definite if and only if it is an extreme character of the infinite hyperoctahedralgroup and we classify the correspondingset of parameters q+, q−. We construct the corresponding representations through a natural action of the hyperoctahedral group B(n) on the tensor product of n copies of a vector space, which gives a two-parameter analog of the classical construction of Schur–Weyl. We apply our classification to construct a cyclic Fock space of type B generalizing the one- parameter construction in type A found previously by Bozejko˙ and Guta. We also construct a new Gaussian operator acting on the cyclic Fock space of type B and we relate its moments with the Askey–Wimp–Kerov distribution by using the notion of cycles on pair-partitions, which we introduce here. Finally, we explain how to solve the analogous problem for the Coxeter groups of type D by using our main result. 1. INTRODUCTION Positive definite functions on a group G play a prominent role in harmonic analysis, opera- tor theory, free probability and geometric group theory. When G has a particularly nice struc- ture of combinatorial/geometric origin one can use it to construct positive definite functions which leads to many interesting properties widely applied in these fields. -
Cluster Algebras and Classical Invariant Rings
Cluster algebras and classical invariant rings by Kevin Carde A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in the University of Michigan 2014 Doctoral Committee: Professor Sergey Fomin, Chair Professor Gordon Belot, Professor Thomas Lam, Professor Karen E. Smith, Professor David E. Speyer ACKNOWLEDGEMENTS I would first like to thank my advisor, Sergey Fomin, for his support these past several years. Besides introducing me to this beautiful corner of combinatorics and providing mathematical guidance throughout, he has been unfailingly supportive even of my pursuits outside of research. I am tremendously grateful on both counts. I would also like to thank David Speyer, whose insight and interest in my work has been one of the best parts of my time at the University of Michigan. I am also grateful to him for suggesting to me a much shorter proof of the irreducibility of mixed invariants. His suggestion, rather than my longer, much less elegant original effort, appears as the end of the proof of Proposition 4.1. Besides these two major mathematical inspirations, I would like to thank the University of Michigan math department at large. From the wonderful people of the graduate office whose patience and care kept me on track on a number of ad- ministrative matters, to the precalculus and calculus coordinators I've worked with who helped spark in me a love of teaching, to the professors whose friendliness and enthusiasm come through both in and out of the classroom, to the graduate students who are always willing to work together through difficult problems (mathematical or otherwise!), everyone I've met here has contributed to the warm, collaborative environment of the department.