Andrei Zelevinsky

Publications

Books

B1. Representations of Finite Classical Groups. A Hopf Algebra Approach 
  Lecture Notes in Mathematics, 869. Springer-Verlag, Berlin-New York, 1981, 184 pp.
 
B2. Discriminants, Resultants and Multidimensional Determinants (with I. M. Gelfand and M. M. Kapranov) 
  Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, 1994, 523 pp.
 

Principal Mathematical Papers

1. Generalized Radon transform for Grassmannians over finite fields (Russian) 
  Uspehi Mat. Nauk 28 (1973), no. 5(173), 243--244. 
 
2. Integral geometry over a finite field (Russian) (with A.B.Zhornitsky) 
  Uspehi Mat. Nauk 28 (1973), no. 6(174), 207--208.
 
3. Representations of the group $SL(2,F_q)$, where $q=2^n$ (Russian) (with G.Narkounskaia) 
  Funkc. Anal. Pril. 8 (1974), no. 3, 75--76.
 
4. Induced representations of reductive ${\germ p}$-adic groups. I (with I.N.Bernstein) 
  Ann. Sci. Éc. Norm. Sup. (4) 10 (1977), no. 4, 441--472.
 
5. Induced representations of reductive ${\germ p}$-adic groups. II.
On irreducible representations of $GL(n)$
  Ann. Sci. Éc. Norm. Sup. (4) 13 (1980), no. 2, 165--210.
 
6. A generalization of the Littlewood-Richardson rule and 
the Robinson-Schensted-Knuth correspondence
  J. Algebra  69 (1981), no. 1, 82--94.
 
7. A $p$-adic analogue of the Kazhdan-Lusztig conjecture (Russian) 
  Funkc. Anal. Pril. 15 (1981), no. 2, 9--21.
 
8. Small resolutions of singularities of Schubert varieties in Grassmannians (Russian) 
  Funkc. Anal. Pril. 17 (1983), no. 2, 75--77.
 
9. Characters of $GL(n, F_q)$ and Hopf algebras (with T.A.Springer) 
  J. London Math. Soc. (2) 30 (1984), no. 1, 27--43.
 
10. Representation models of classical groups and their hidden symmetries (Russian) (with I.M.Gelfand) 
  Funkc. Anal. Pril. 18 (1984), no. 3, 14--31.
 
11. Two remarks on graded nilpotent classes (Russian) 
  Uspehi Mat. Nauk 40 (1985), no. 1(241), 199--200. 
 
12. Representation models for classical groups and their higher symmetries (with I.M.Gelfand) 
  Astérisque1985, The mathematical heritage of Élie Cartan (Lyon, 1984), Numero Hors Serie, 117--128.
 
13. Multiplicities and good bases for $gl_n$ (with I.M.Gelfand) 
  Group theoretical methods in physics, Vol. II (Yurmala, 1985), 147--159, VNU Sci. Press, Utrecht, 1986.
 
14. Canonical basis in irreducible representations of $gl_3$ and its applications (with I.M.Gelfand) 
  Group theoretical methods in physics, Vol. II (Yurmala, 1985), 127--146, VNU Sci. Press, Utrecht, 1986.
 
15. Algebraic and combinatorial aspects of the general theory of hypergeometric functions (Russian) (with I.M.Gelfand) 
  Funkc. Anal. Pril. 20 (1986), no. 3, 17--34.
 
16. General hypergeometric functions on complex Grassmannians (Russian) (with V.A.Vasiliev and I.M.Gelfand) 
  Funkc. Anal. Pril. 21 (1987), no. 1, 23--38.
 
17. Resolutions, dual pairs, and character formulas (Russian) 
  Funkc. Anal. Pril. 21 (1987), no. 2, 74--75.
 
18. Holonomic systems of equations and series of hypergeometric type (Russian) (with I.M.Gelfand and M.I.Graev) 
  Dokl. Akad. Nauk SSSR 295 (1987), no. 1, 14--19; translation in Soviet Math. Dokl. 36 (1988), no. 1, 5--10.
 
19. Arrangements of real hyperplanes and related partition function (Russian) (with T.V.Alekseevskaya and I.M.Gelfand) 
  Dokl. Akad. Nauk SSSR 297 (1987), no. 6, 1289--1293; translation in Soviet Math. Dokl. 36 (1988), no. 3, 589--593.
 
20. The base affine space and canonical bases in irreducible representations of the group $Sp(4)$ (Russian) (with V.S.Retakh ) 
  Dokl. Akad. Nauk SSSR 300 (1988), no. 1, 31--35; translation in Soviet Math. Dokl. 37 (1988), no. 3, 618--622.
 
21. Involutions on Gelfand-Tsetlin patterns and multiplicities in skew $GL(n)$-modules (Russian) (with A.D.Berenstein) 
  Dokl. Akad. Nauk SSSR 300 (1988), no. 6, 1291--1294; translation in Soviet Math. Dokl. 37 (1988), no. 3, 799--802.
 
22. Hypergeometric functions and toric varieties (Russian) (with I.M.Gelfand and M.M.Kapranov) 
  Funkc. Anal. Pril. 23 (1989), no. 2, 12--26; translation in Funct. Anal. Appl. 23 (1989), no. 2, 94--106 
 
23. General hypergeometric functions associated to a pair of homogeneous spaces (Russian) (with I.M.Gelfand and V.V.Serganova) 
  Dokl. Akad. Nauk SSSR 304 (1989), no. 5, 1044--1049; translation in Soviet Math. Dokl. 39 (1989), no. 1, 182--187.
 
24. Projectively dual varieties and hyperdeterminants (Russian) (with I. M. Gelfand and M. M. Kapranov) 
  Dokl. Akad. Nauk SSSR 305 (1989), no. 6, 1294--1298; translation in Soviet Math. Dokl. 39 (1989), no. 2, 385--389.
 
25. Combinatorial optimization on Weyl groups, greedy algorithms, and generalized matroids (Russian) (with V.V.Serganova) 
  Preprint, Scientific Council for Cybernetics, Moscow 1989.
 
26. Geometry and combinatorics related to vector partition functions
  Topics in algebra, Part 2 (Warsaw, 1988), 501--510, Banach Center Publ., 26, Part 2, PWN, Warsaw, 1990. 
 
27. Tensor product multiplicities and convex polytopes in partition space (with A.D.Berenstein) 
  J. Geom. Phys. 5 (1988), no. 3, 453--472.
 
28. On discriminants of polynomials of several variables (Russian) (with I.M.Gelfand and M.M.Kapranov) 
  Funkc. Anal. Pril. 24 (1990), no. 1, 1--4; translation in Funct. Anal. Appl. 24 (1990), no. 1, 1--4 
 
29. Discriminants of polynomials of several variables and triangulations of Newton polytopes (Russian) (with I.M.Gelfand and M.M.Kapranov) 
  Algebra i Analiz 2 (1990), no. 3, 1--62; translation in Leningrad Math. J. 2 (1991), no. 3, 449--505.
 
30. When is the weight multiplicity equal to $1$ (Russian) (with A.D.Berenstein) 
  Funkc. Anal. Pril. 24 (1990), no. 4, 1--13; translation in Funct. Anal. Appl. 24 (1990), no. 4, 259--269. 
 
31. Newton polytopes of the classical resultant and discriminant (with I. M. Gelfand and M. M. Kapranov) 
  Adv. Math. 84 (1990), no. 2, 237--254. 
 
32. Generalized Euler integrals and $A$-hypergeometric functions (with I. M. Gelfand and M. M. Kapranov) 
  Adv. Math. 84 (1990), no. 2, 255--271. 
 
33. Quotients of toric varieties (with M.M.Kapranov and B.Sturmfels) 
  Math. Ann. 290 (1991), no. 4, 643--655.
 
34. Hypergeometric functions, toric varieties, and Newton polyhedra (with I.M.Gelfand and M.M.Kapranov) 
  Special functions (Okayama, 1990), 104--121, ICM-90 Satell. Conf. Proc., Springer, Tokyo, 1991.
 
35. Triple multiplicities for $sl(r+1)$ and the spectrum of the exterior algebra of the adjoint representation (with A. D. Berenstein) 
  J. Algebraic Combin. 1 (1992), no. 1, 7--22.
 
36. Chow polytopes and general resultants (with M.M.Kapranov and B.Sturmfels) 
  Duke Math. J. 67 (1992), no. 1, 189--218.
 
37. Hyperdeterminants (with I. M. Gelfand and M. M. Kapranov) 
  Adv. Math. 96 (1992), no. 2, 226--263. 
 
38. Maximal minors and their leading terms (with Bernd Sturmfels) 
  Adv. Math. 98 (1993), no. 1, 65--112. 
 
39. Combinatorics of maximal minors (with David Bernstein) 
  J. Algebraic Combin. 2 (1993), no. 2, 111--121.
 
40. String bases for quantum groups of type $A_r$ (with Arkady Berenstein) 
  I. M. Gelfand Seminar, 51--89, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc., Providence, RI, 1993.
 
41. Multigraded resultants of Sylvester type (with Bernd Sturmfels) 
  J. Algebra 163 (1994), no. 1, 115--127.
 
42. Simple vertices of maximal minor polytopes (with Prakash Santhanakrishnan) 
  Discrete Comput. Geom. 11 (1994), no. 3, 289--309. 
 
43. Determinantal formulas for multigraded resultants (with Jerzy Weyman) 
  J. Algebraic Geom. 3 (1994), no. 4, 569--597. 
 
44. Multiplicative properties of projectively dual varieties (with Jerzy Weyman) 
  Manuscripta Math. 82 (1994), no. 2, 139--148.
 
45. Representations of quivers of type $A$ and the multisegment duality (with Harold Knight) 
  Adv. Math.  117 (1996), no. 2, 273--293. 
 
46. Canonical bases for the quantum group of type $A_r$ and piecewise-linear combinatorics (with Arkady Berenstein) 
  Duke Math. J. 82 (1996), no. 3, 473--502.
 
47. Singularities of hyperdeterminants (with J. Weyman) 
  Ann. Inst. Fourier (Grenoble) 46 (1996), no. 3, 591--644.
 
48. Parametrizations of canonical bases and totally positive matrices (with Arkady Berenstein and Sergey Fomin) 
  Adv. Math. 122 (1996), no. 1, 49--149. 
 
49. Total positivity in Schubert varieties (with Arkady Berenstein) 
  Comment. Math. Helv. 72 (1997), no. 1, 128--166.
 
50. Polyhedral realizations of crystal bases for quantized Kac-Moody algebras (with Toshiki Nakashima) 
  Adv. Math. 131 (1997), no. 1, 253--278. 
 
51. A geometric characterization of Coxeter matroids (with V.V.Serganova and A.Vince) 
  Annals of Combin. 1 (1997), no. 1, 173--181. 
 
52. Quasicommuting families of quantum Plucker coordinates (with Bernard Leclerc) 
  Amer. Math. Soc. Transl. (2) Vol. 181, Kirillov's Seminar on Representation Theory, 85--108, Amer. Math. Soc., Providence, RI, 1998.
 
53. Double Bruhat cells and total positivity (with Sergey Fomin) 
  J. Amer. Math. Soc.  12 (1999), no. 2, 335--380 .
 
54. Multiple flag varieties of finite type (with Peter Magyar and Jerzy Weyman) 
  Adv. Math. 141 (1999), no. 1, 97--118. 
 
55. Symplectic multiple flag varieties of finite type (with Peter Magyar and Jerzy Weyman) 
  J. Algebra 230 (2000), no. 1, 245--265. 
 
56. Recognizing Schubert cells (with Sergey Fomin) 
  J. Algebraic Combin.  12 (2000) 37--57. 
 
57. Totally nonnegative and oscillatory elements in semisimple groups (with Sergey Fomin) 
  Proc. Amer. Math. Soc. 128 (2000), no. 12,  3749-3759.
 
58. Multiplicities of points on Schubert varieties in Grassmannians (with Joachim Rosenthal) 
  J. Algebraic Combin.  13 (2001), no. 2,  213 -- 218.
 
59. Simply-laced Coxeter groups and groups generated by symplectic transvections (with Boris Shapiro, Michael Shapiro and Alek Vainshtein) 
  Michigan Math. J.   48  (2000), 531-551.
 
60. Tensor product multiplicities, canonical bases and totally positive varieties (with Arkady Berenstein) 
  Invent. math. 143  (2001), no.1,  77-128.
 
61. Connected components of real double Bruhat cells
  Intern. Math. Res. Notices 2000, No. 21, 1131- 1153.
 
62. Cluster algebras I: Foundations (with Sergey Fomin)
  J. Amer. Math. Soc. 15 (2002), no. 2, 497--529.
 
63. The Laurent phenomenon (with Sergey Fomin) 
  Adv. Applied Math.  28 (2002), no. 2, 119--144.
 
64. Y-systems and generalized associahedra (with Sergey Fomin) 
  Annals of Mathematics.  158 (2003), No. 3,  977--1018.
 
65. Polytopal realizations of generalized associahedra (with Frederic Chapoton and Sergey Fomin) 
  Canadian Math. Bulletin.  45 (2002), no. 4, 537--566.
 
66. On symplectic leaves and integrable systems in standard complex semisimple Poisson-Lie groups (with Mikhail Kogan ) 
  Intern. Math. Res. Notices 2002, No.32, 1685-1702.
 
67. Generalized associahedra via quiver representations (with Robert Marsh and Markus Reineke) 
  Trans. Amer. Math. Soc.,  355 (2003),  No. 10, 4171-4186 . 
 
68. Cluster algebras II: Finite type classification (with Sergey Fomin) 
  Invent. math., 154 (2003), no.1, 63-121.
 
69. Cluster algebras III: Upper bounds and double Bruhat cells (with Arkady Berenstein and Sergey Fomin) 
  Duke Math. J.   126 (2005),  No. 1, 1–52.
 
70. Positivity and canonical bases in rank 2 cluster algebras of finite and affine types (with Paul Sherman) 
  Moscow Math. J.  4 (2004),  No. 4, 947-974 . 
 
71. Quantum cluster algebras (with Arkady Berenstein) 
  Advances in Math.   195 (2005),  No. 2, 405-455. 
 
72. Cluster algebras of finite type and positive symmetrizable matrices (with Michael Barot and Christof Geiss) 
  J. London Math. Soc.   73 (2006),  Part 3, 545–564. 
 
73. Nested complexes and their polyhedral realizations  
  Pure and Applied Mathematics Quarterly   2 (2006),  No. 3, 655-671. 
 
74. Cluster algebras IV: Coefficients (with Sergey Fomin) 
  Compositio Mathematica   143 (2007),  112-164. 
 
75. Laurent expansions in cluster algebras via quiver representations (with Philippe Caldero) 
  Moscow Math. J.  6 (2006),  No. 3, 411-429 . 
 
76. Semicanonical basis generators of the cluster algebra of type $A_1^{(1)}$ 
  Electron. J. Combin.  14 (2007),  N4. 
 
77. Quivers with potentials and their representations I: Mutations (with Harm Derksen and Jerzy Weyman) 
  arXiv:0704.0649 [math.RA],  April 18, 2007, 58 pp.; to appear in Selecta Mathematica.
 
78. Cluster algebras of finite type via Coxeter elements and principal minors (with Shih-Wei Yang) 
  arXiv:0804.3303 [math.RA],  April 21, 2008, 38 pp.
 

Surveys and Expository Articles

S1. Representations of the group $GL(n,F),$ where $F$ is a local non-Archimedean field (Russian) (with J.N.Bernstein) 
  Uspehi Mat. Nauk 31 (1976), no. 3(189), 5--70. 
 
S2. Representations of contragredient Lie algebras and the Kac-Macdonald identities (with B.L.Feigin) 
  Representations of Lie groups and Lie algebras (Budapest, 1971), 25--77, Akad. Kiadó, Budapest, 1985.
 
S3. Chow polytopes
  Special Differential Equations, Proceedings of the Taniguchi workshop 1991, M.Yoshida (Ed.) 176--181, Department of Mathematics, Kyushu University, Fukuoka 812, Japan, 1991.
 
S4. Littlewood-Richardson semigroups
  New Perspectives in Algebraic Combinatorics, MSRI Publications, Vol. 38, 337 -- 345, Cambridge University Press, 1999.
 
S5. Multisegment duality, canonical bases and total positivity
  Documenta Mathematica, Extra Volume ICM 1998, III , 409--417.
 
S6. Total positivity: tests and parametrizations(with Sergey Fomin) 
  Mathematical Intelligencer,   22 (2000), no. 1, 23--33. 
 
S7. From Littlewood-Richardson coefficients to cluster algebras in three lectures 
  Symmetric Functions 2001: Surveys of Developments and Perspectives, Proceedings of the NATO Advanced Study Institute 2001,  S.Fomin (Ed.), 253-273;  NATO Science Series II: Mathematics, Physics and Chemistry, Vol. 74. Kluwer Academic Publishers, Dordrecht, 2002.
 
S8. Cluster algebras: Notes for the CDM-03 conference (with Sergey Fomin) 
  CDM 2003: Current Developments in Mathematics.  International Press, 2004.
 
S9. Cluster algebras: origins, results and conjectures  
  Advances in Algebra towards Millenium Problems, Proceedings of 2004 International Conference on Related Subjects to Clay Problems,  Ki-Bong Nam et al. (Eds.), 85-105; SAS International Publications, Delhi, 2005.
 
S10. Quantum cluster algebras: Oberwolfach talk, February 2005  
  ArXiv:math.QA/0502260 February 13, 2005, 3 pages.  
 
S11. Generalized Littlewood-Richardson coefficients, canonical bases and total positivity (Russian) 
  Globus. General mathematical seminar,   M.A.Tsfasman (Ed.), 147-160; Independent Moscow University, Moscow, 2004.
 
S12. Mutations for quivers with potentials: Oberwolfach talk, April 2007  
  arXiv:0706.0822 [math.RA], June 6, 2007, 4 pages.  
 
S13. What is ... a Cluster Algebra,  
  Notices of the AMS,   54 (2007), no. 11, 1494--1495.  
 

Research Announcements

A1. Classification of irreducible noncuspidal representations of $GL(n)$ over a $p$-adic field (Russian) 
  Funkc. Anal. Pril. 11 (1977), no. 1, 67--68.
 
A2. Representation ring of the group $GL(n)$ over a $p$-adic field (Russian) 
  Funkc. Anal. Pril. 11 (1977), no. 3, 78--79.
 
A3. Representations of the general linear and affine groups over a finite field (Russian) 
  Uspehi Mat. Nauk 32 (1977), no. 3(195), 159--160. 
 
A4. Convex polytopes in the pattern space and canonical basis in irreducible representations of $gl(3)$ (Russian) (with I. M. Gelfand) 
  Funkc. Anal. Pril. 19 (1985), no. 2, 72--75.
 
A5. The behavior of general hypergeometric functions in a complex domain (Russian) (with I.M.Gelfand and V.A.Vasiliev) 
  Dokl. Akad. Nauk SSSR 290 (1986), no. 2, 277--281. 
 
A6. Equations of hypergeometric type and Newton polytopes (Russian) (with I.M.Gelfand and M.M.Kapranov) 
  Dokl. Akad. Nauk SSSR 300 (1988), no. 3, 529--534; translation in Soviet Math. Dokl. 37 (1988), no. 3, 678--682. 
 
A7. $A$-discriminants and Cayley-Koszul complexes (Russian) (with I.M.Gelfand and M.M.Kapranov) 
  Dokl. Akad. Nauk SSSR 307 (1989), no. 6, 1307--1311; translation in Soviet Math. Dokl. 40 (1990), no. 1, 239--243. 
 
A8. Newton polytopes of principal $A$-determinants (Russian) (with I.M.Gelfand and M.M.Kapranov) 
  Dokl. Akad. Nauk SSSR 308 (1989), no. 1, 20--23; translation in Soviet Math. Dokl. 40 (1990), no. 2, 278--281. 
 
A9. Multiple flag varieties of finite type (with Peter Magyar and Jerzy Weyman) 
  Commutative algebra, representation theory and combinatorics, D.Eisenbud, A.Martsinkovsky, J.Weyman (Eds.), Conference in honor of David Buchsbaum, 115--119, Northeastern University, Boston, 1997.
 

Papers in Computational Seismology

CS1. The study of sites of strongest earthquakes at the Pacific Belt by pattern recognition algorithms (Russian)(with A.D.Gvishiani, V.I.Keilis-Borok and V.G.Kosobokov) 
  Izvestiya Akad. Nauk SSSR, Earth Physics 9 (1978), 31--42. 
 
CS2. Recognition of sites of strongest earthquakes at the Pacific Belt ($M \ge 8.2$) (Russian)(with A.D.Gvishiani, V.I.Keilis-Borok and V.G.Kosobokov) 
  Computational Seismology 13 , 30--43, Nauka, Moscow, 1980.
 
CS3. Recognition of possible sites of strong earthquakes XI. Western Alps, $M \ge 5.0$ (Russian)(with A.I.Gorshkov and E.Ya.Rantsman) 
  Computational Seismology 15 , 67--73, Nauka, Moscow, 1983.
 
CS4. On a percolation model of seismicity (Russian) 
  Computational Seismology 18 , 10--24, Nauka, Moscow, 1985.
 

Papers in Mathematical Education

E1. Chebyshev polynomials and recursive relations (Russian)(with N.B.Vasiliev) 
  Kvant  (1978), no. 1, 12--19; translation in Quantum   10 (1999), no. 1, 20--26.
 
E2. Young diagrams as research material for high school students (Russian) 
  All-Union Correspondence Math. School, 46--51, Moscow, 1984.
 
E3. Visibles revisited (with Mark Bridger) 
  College Math. J.  36 (2005),  No. 4, 289-300. 
 

Miscellaneous

M1. Remembering Bella Abramovna ( in Russian)
  Mathematical Education (Matematicheskoe prosveshchenie).  Ser. 3, No. 9 (2005),  22--29;
English translation in: You failed your math test, Comrade Einstein,   M.Shifman (Ed.), 191-195;  World Scientific, 2005.
 
M2. Interview for in-cites, 
  Thomson Scientific,  March 2006. 
 
Home page: Andrei Zelevinsky Started: April 10, 1998.  Last modified: April   21, 2008
Email:  andrei (at) neu (dot) edu URL: http://www.math.neu.edu/zelevinsky/publications.html