The Invariant Theory of Matrices

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UNIVERSITY LECTURE SERIES VOLUME 69

Corrado De Concini Claudio Procesi

American Mathematical Society The Invariant Theory of Matrices

10.1090/ulect/069

UNIVERSITY LECTURE SERIES VOLUME 69

The Invariant Theory of Matrices

Corrado De Concini Claudio Procesi

American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Jordan S. Ellenberg Robert Guralnick William P. Minicozzi II (Chair) Tatiana Toro

2010 Mathematics Subject Classiﬁcation. Primary 15A72, 14L99, 20G20, 20G05.

For additional information and updates on this book, visit www.ams.org/bookpages/ulect-69

Library of Congress Cataloging-in-Publication Data Names: De Concini, Corrado, author. | Procesi, Claudio, author. Title: The invariant theory of matrices / Corrado De Concini, Claudio Procesi. Description: Providence, Rhode Island : American Mathematical Society, [2017] | Series: Univer- sity lecture series ; volume 69 | Includes bibliographical references and index. Identiﬁers: LCCN 2017041943 | ISBN 9781470441876 (alk. paper) Subjects: LCSH: Matrices. | Invariants. | AMS: Linear and multilinear algebra; matrix theory – Basic linear algebra – Vector and tensor algebra, theory of invariants. msc | Algebraic geometry – Algebraic groups – None of the above, but in this section. msc | Group theory and generalizations – Linear algebraic groups and related topics – Linear algebraic groups over the reals, the complexes, the quaternions. msc | Group theory and generalizations – Linear algebraic groups and related topics – Representation theory. msc Classiﬁcation: LCC QA188 .D425 2017 | DDC 512.9/434–dc23 LC record available at https://lccn. loc.gov/2017041943

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Introductionandpreliminaries ...... 1 1. Introduction...... 2 2. Preliminaries...... 8

PartI. Theclassicaltheory...... 19 3. Representationtheory...... 20 4. Algebraswithtrace...... 28

PartII. Quasi-hereditaryalgebras...... 39 5. Modules...... 40 6. Goodﬁltrationsandquasi-hereditaryalgebras...... 43

PartIII. TheSchuralgebra...... 49 7. TheSchuralgebra...... 50 8. Doubletableaux...... 51 9. ModulesfortheSchuralgebra...... 62 10. Rational GL(m)-modules...... 75 11. Tensorproducts...... 78

PartIV. Matrixfunctionsandinvariants...... 87 12. Areductionforinvariantsofseveralmatrices ...... 88 13. Polarizationandspecialization...... 91 14. Exteriorproducts...... 95 15. Matrixfunctionsandinvariants...... 99

PartV. Relations...... 107 16. Relations...... 108 17. Describing Km ...... 110 18. Km versus K˜m ...... 118

PartVI. TheSchuralgebraofafreealgebra...... 131 19. Preliminaryfacts ...... 132 20. TheSchuralgebraofthefreealgebra...... 135 Bibliography...... 145 GeneralIndex...... 149 SymbolIndex...... 151

Bibliography

[1]E.Aljadeff,A.Giambruno,C.Procesi,A.Regev,Rings with polynomial identities and finite dimensional representations of algebras, book in preparation. [2] S. A. Amitsur, On the characteristic polynomial of a sum of matrices, Linear and Multilinear Algebra 8 (1979/80), no. 3, 177–182, DOI 10.1080/03081088008817315. MR560557 [3] H. Aslaksen, E.-C. Tan, and C.-b. Zhu, Invariant theory of special orthogonal groups,Pacific J. Math. 168 (1995), no. 2, 207–215. MR1339950 [4] I. Assem, D. Simson, and A. Skowroński, Elements of the representation theory of associative algebras. Vol. 1. Techniques of representation theory, London Mathematical Society Student Texts, vol. 65, Cambridge University Press, Cambridge, 2006. MR2197389 [5] A. Bergmann, Formen auf Modulnuber ¨ kommutativen Ringen beliebiger Charakteristik (German), J. Reine Angew. Math. 219 (1965), 113–156, DOI 10.1515/crll.1965.219.113. MR0190167 [6] A. Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer- Verlag, New York, 1991. MR1102012 [7] A. Borel, Density properties for certain subgroups of semi-simple groups without compact components, Ann. of Math. (2) 72 (1960), 179–188, DOI 10.2307/1970150. MR0123639 [8] R. Bott, Homogeneous vector bundles, Ann. of Math. (2) 66 (1957), 203–248, DOI 10.2307/1969996. MR0089473 [9] E. Cline, B. Parshall, and L. Scott, Algebraic stratification in representation categories,J. Algebra 117 (1988), no. 2, 504–521, DOI 10.1016/0021-8693(88)90123-8. MR957457 [10] E. Cline, B. Parshall, and L. Scott, Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85–99. MR961165 [11] E. Cline, B. Parshall, and L. Scott, Integral and graded quasi-hereditary algebras. I,J.Algebra 131 (1990), no. 1, 126–160, DOI 10.1016/0021-8693(90)90169-O. MR1055002 [12] E. Cline, B. Parshall, and L. Scott, The homological dual of a highest weight category,Proc. London Math. Soc. (3) 68 (1994), no. 2, 294–316, DOI 10.1112/plms/s3-68.2.294. MR1253506 [13] E. Cline, B. Parshall, and L. Scott, Stratifying endomorphism algebras,Mem.Amer.Math. Soc. 124 (1996), no. 591, viii+119, DOI 10.1090/memo/0591. MR1350891 [14] C. de Concini, D. Eisenbud, and C. Procesi, Young diagrams and determinantal varieties, Invent. Math. 56 (1980), no. 2, 129–165, DOI 10.1007/BF01392548. MR558865 [15] V. Dlab and C. M. Ringel, Quasi-hereditary algebras, Illinois J. Math. 33 (1989), no. 2, 280–291. MR987824 [16] V. Dlab, P. Heath, and F. Marko, Quasi-heredity of endomorphism algebras,C.R.Math. Rep. Acad. Sci. Canada 16 (1994), no. 6, 277–282. MR1321690 [17] M. Domokos, S. G. Kuzmin, and A. N. Zubkov, Rings of matrix invariants in positive characteristic, J. Pure Appl. Algebra 176 (2002), no. 1, 61–80, DOI 10.1016/S0022-4049(02)00117-2. MR1930966 [18] M. Domokos and P. E. Frenkel, Mod 2 indecomposable orthogonal invariants, Adv. Math. 192 (2005), no. 1, 209–217, DOI 10.1016/j.aim.2004.04.005. MR2122284 [19] S. Donkin, Rational representations of algebraic groups, Lecture Notes in Mathematics, vol. 1140, Springer-Verlag, Berlin, 1985. Tensor products and filtration. MR804233 [20] S. Donkin, Invariants of several matrices, Invent. Math. 110 (1992), no. 2, 389–401, DOI 10.1007/BF01231338. MR1185589 [21] S. Donkin, Invariant functions on matrices, Math. Proc. Cambridge Philos. Soc. 113 (1993), no. 1, 23–43, DOI 10.1017/S0305004100075757. MR1188816 [22] P. Doubilet, G.-C. Rota, and J. Stein, On the foundations of combinatorial theory. IX. Combi- natorial methods in invariant theory, Studies in Appl. Math. 53 (1974), 185–216. MR0498650

145 146 BIBLIOGRAPHY

[23] V. Drensky, Computing with matrix invariants, Math. Balkanica (N.S.) 21 (2007), no. 1-2, 141–172. MR2350726 [24] E. Formanek, The invariants of n × n matrices, Invariant theory, Lecture Notes in Math., vol. 1278, Springer, Berlin, 1987, pp. 18–43, DOI 10.1007/BFb0078804. MR924163 [25] E. Formanek, The polynomial identities and invariants of n × n matrices, CBMS Regional Conference Series in Mathematics, vol. 78, Published for the Conference Board of the Math- ematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1991. MR1088481 [26] W. Fulton and J. Harris, Representation theory: A first course, Readings in Mathematics. Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. MR1153249 [27] J. A. Green, Polynomial representations of GLn, Lecture Notes in Mathematics, vol. 830, Springer-Verlag, Berlin-New York, 1980. MR606556 [28] M. Hall Jr., Combinatorial theory, 2nd ed., Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Inc., New York, 1986. A Wiley-Interscience Publication. MR840216 [29] W. V. D. Hodge, Some enumerative results in the theory of forms, Proc. Cambridge Philos. Soc. 39 (1943), 22–30. MR0007739 [30] J. E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978. Second printing, revised. MR499562 [31] J.-i. Igusa, On the arithmetic normality of the Grassmann variety,Proc.Nat.Acad.Sci.U. S. A. 40 (1954), 309–313. MR0061423 [32] G. D. James, The representation theory of the symmetric groups, Lecture Notes in Mathe- matics, vol. 682, Springer, Berlin, 1978. MR513828 [33] G. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by P. M. Cohn; With an introduction by Gilbert de B. Robinson. MR644144 [34] J. C. Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Mono- graphs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR2015057 [35] B. Kostant, A theorem of Frobenius, a theorem of Amitsur-Levitski and cohomology theory, J. Math. Mech. 7 (1958), 237–264. MR0092755 [36] V. Lakshmibai and C. S. Seshadri, Geometry of G/P .II.TheworkofdeConciniandProcesi and the basic conjectures, Proc. Indian Acad. Sci. Sect. A 87 (1978), no. 2, 1–54. MR490244 [37] A. Lascoux, Syzygies des variétés déterminantales (French), Adv. in Math. 30 (1978), no. 3, 202–237, DOI 10.1016/0001-8708(78)90037-3. MR520233 [38] J. S. Lew, The generalized Cayley-Hamilton theorem in n dimensions (English, with French summary), Z. Angew. Math. Phys. 17 (1966), 650–653, DOI 10.1007/BF01597249. MR0213381 [39] A. A. Lopatin, Relations between O(n)-invariants of several matrices, Algebr. Represent. Theory 15 (2012), no. 5, 855–882, DOI 10.1007/s10468-011-9268-4. MR2969280 [40] M. Lothaire, Combinatorics on words, Cambridge Mathematical Library, Cambridge Univer- sity Press, Cambridge, 1997. With a foreword by Roger Lyndon and a preface by Dominique Perrin; Corrected reprint of the 1983 original, with a new preface by Perrin. MR1475463 [41] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contri- butions by A. Zelevinsky; Oxford Science Publications. MR1354144 [42] O. Mathieu, Filtrations of G-modules, Ann. Sci. Ecole´ Norm. Sup. (4) 23 (1990), no. 4, 625–644. MR1072820 [43] V. B. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. of Math. (2) 122 (1985), no. 1, 27–40, DOI 10.2307/1971368. MR799251 [44] C. Procesi, The invariants of n × n matrices, Bull. Amer. Math. Soc. 82 (1976), no. 6, 891–892, DOI 10.1090/S0002-9904-1976-14196-1. MR0419490 [45] C. Procesi, The invariant theory of n × n matrices, Advances in Math. 19 (1976), no. 3, 306–381, DOI 10.1016/0001-8708(76)90027-X. MR0419491 [46] C. Procesi, Trace identities and standard diagrams, Ring theory (Proc. Antwerp Conf. (NATO Adv. Study Inst.), Univ. Antwerp, Antwerp, 1978), Lecture Notes in Pure and Appl. Math., vol. 51, Dekker, New York, 1979, pp. 191–218. MR563295 BIBLIOGRAPHY 147

[47] C. Procesi, A formal inverse to the Cayley-Hamilton theorem,J.Algebra107 (1987), no. 1, 63–74, DOI 10.1016/0021-8693(87)90073-1. MR883869 [48] C. Procesi, Lie groups. An approach through invariants and representations,Universitext, Springer, New York, 2007. MR2265844 [49] S. Ramanan and A. Ramanathan, Projective normality of flag varieties and Schubert varieties, Invent. Math. 79 (1985), no. 2, 217–224, DOI 10.1007/BF01388970. MR778124 [50] J. Razmyslov, Trace identities of matrix algebras via a field of characteristic zero,Math. USSR Izvestia (translation) 8 (1974), pp. 727–760. [51] Ju. P. Razmyslov, A certain problem of Kaplansky (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 483–501. MR0338063 [52] Yu. P. Razmyslov, Trace identities of full matrix algebras over a field of characteristic zero, Math. USSR Izv. 8 (1974), pp. 724-760. [53] C. Reutenauer and M.-P. Schützenberger, A formula for the determinant of a sum of matrices, Lett. Math. Phys. 13 (1987), no. 4, 299–302, DOI 10.1007/BF00401158. MR895292 [54] N. Roby, Loispolynomesetloisformellesenthéorie des modules (French), Ann. Sci. Ecole´ Norm. Sup. (3) 80 (1963), 213–348. MR0161887 [55] N. Roby, Lois polynômes multiplicatives universelles (French, with English summary), C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 19, A869–A871. MR580160 [56] B. E. Sagan, The symmetric group. Representations, combinatorial algorithms, and symmetric functions, 2nd ed., Graduate Texts in Mathematics, vol. 203, Springer-Verlag, New York, 2001. MR1824028 [57] I. Schur, Uber¨ eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen, Diss. Berlin (1901). [58] C. S. Seshadri, Geometry of G/P . I. Theory of standard monomials for minuscule representations, C. P. Ramanujam—a tribute, Tata Inst. Fund. Res. Studies in Math., vol. 8, Springer, Berlin-New York, 1978, pp. 207–239. MR541023 [59] K. S. Sibirski˘ı, Algebraic invariants of a system of matrices (Russian), Sibirsk. Mat. Z.ˇ 9 (1968), 152–164. MR0223379 [60] A. J. M. Spencer and R. S. Rivlin, The theory of matrix polynomials and its application to the mechanics of isotropic continua, Arch. Rational Mech. Anal. 2 (1958/1959), 309–336, DOI 10.1007/BF00277933. MR0100600 [61] A. J. M. Spencer and R. S. Rivlin, Finite integrity bases for five or fewer symmetric 3 × 3 matrices, Arch. Rational Mech. Anal. 2 (1958/1959), 435–446, DOI 10.1007/BF00277941. MR0100601 [62] A. J. M. Spencer, The invariants of six symmetric 3 × 3 matrices, Arch. Rational Mech. Anal. 7 (1961), 64–77, DOI 10.1007/BF00250750. MR0120246 [63] A. J. M. Spencer and R. S. Rivlin, Further results in the theory of matrix polynomials,Arch. Rational Mech. Anal. 4 (1960), 214–230 (1960), DOI 10.1007/BF00281388. MR0109830 [64] Spencer, A. J. M., Theory of invariants (A. C. Eringen, Editor), Continuum Physics, vol. I, Academic Press, San Diego (1971), pp. 240–353. [65] A. J. M. Spencer, Ronald Rivlin and invariant theory, Internat. J. Engrg. Sci. 47 (2009), no. 11-12, 1066–1078, DOI 10.1016/j.ijengsci.2009.01.004. MR2590475 [66] T. A. Springer, Linear algebraic groups, 2nd ed., Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2009. MR2458469 [67] M. E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR0252485 [68] J. Towber, Young symmetry, the flag manifold, and representations of GL(n), J. Algebra 61 (1979), no. 2, 414–462, DOI 10.1016/0021-8693(79)90289-8. MR559849 [69] F. Vaccarino, Generalized symmetric functions and invariants of matrices,Math.Z.260 (2008), no. 3, 509–526, DOI 10.1007/s00209-007-0285-2. MR2434467 [70] F. Vaccarino, Homogeneous multiplicative polynomial laws are determinants, J. Pure Appl. Algebra 213 (2009), no. 7, 1283–1289, DOI 10.1016/j.jpaa.2008.11.036. MR2497575 [71] J. P. Wang, Sheaf cohomology on G/B and tensor products of Weyl modules,J.Algebra77 (1982), no. 1, 162–185, DOI 10.1016/0021-8693(82)90284-8. MR665171 [72] H. Weyl, The classical groups. Their invariants and representations, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Fifteenth printing; Princeton Paperbacks. MR1488158 148 BIBLIOGRAPHY

[73] D. Ziplies, Generators for the divided powers algebra of an algebra and trace identities, Beitr¨age Algebra Geom. 24 (1987), 9–27. MR888200 [74] D. Ziplies, Abelianizing the divided powers algebra of an algebra,J.Algebra122 (1989), no. 2, 261–274, DOI 10.1016/0021-8693(89)90215-9. MR999072 [75] A. N. Zubkov, Endomorphisms of tensor products of exterior powers and Procesi hypothesis, Comm. Algebra 22 (1994), no. 15, 6385–6399, DOI 10.1080/00927879408825195. MR1303010 [76] A. N. Zubkov, On a generalization of the Razmyslov-Procesi theorem (Russian, with Rus- sian summary), Algebra i Logika 35 (1996), no. 4, 433–457, 498, DOI 10.1007/BF02367026; English transl., Algebra and Logic 35 (1996), no. 4, 241–254. MR1444429 [77] A. N. Zubkov, Invariants of an adjoint action of classical groups (Russian, with Russian summary), Algebra Log. 38 (1999), no. 5, 549–584, 639, DOI 10.1007/BF02671747; English transl., Algebra and Logic 38 (1999), no. 5, 299–318. MR1766702 General Index

Δ-filtration, 44 full, 24 ∇-filtration, 44 polynomial law, 9 projective cover, 40 adjacency, 52 affine scheme, 16 rational representation, 13 group, 16 restitution, 24 algebra rim quasi hereditary, 46 of a diagram, 76 Schur, 50 row bitableau, 59 trace, 28 algebra with trace, 28 Schur functors, 66 Schur module, 65 bitableaux Schur symmetric function, 20 semistandard, 60 shape of the diagram, 51 simple degeneration, 103 canonical tableau, 53 skew diagram, 51 Cayley–Hamilton identity, 30 specialization, 24 composition, 95 Standard filtration, 68 cyclic equivalence, 5 straightening relation, 56 dominance order, 51 trace formal, 28 equivariant maps, 26 essential extension, 40 weight, 51 of a bitableau, 63 free Schur algebra, 137 dominant, 51 full polarization, 92 Weyl module, 69 height, 20 Young diagram, 51 injective hull, 40 Young superclass, 103 Young tableau, 52 linear algebraic group, 12 standard, 52 matrix variable, 34 module socle, 42 superfluous, 40 tilting, 82 top, 42 monomial primitive, 4 multilinearization, 24 multiplicative map, 11 partial polarization, 92 polarization

149

Symbol Index

SA = SAX,37 σi,34 Tm,38 Q˜λ := Rm,t(λ), 72 Tst ⊂ T λ λ,88 εj1,...,jk ,9 A[G], 12 |λ|,51 Dμ,99 bj , 125 F+X,5 e(λ), 104 I α L ⊂∇λ,73 e = e ,10 f Mλ,88 e = σf1,...,fh (M1,...,Mh), 138 ∗ R ⊂ F [xi,j ], 56 f ,40 Rn,50 ht(λ), 20 → SCg, 103 k :[1,...,N] [1,...,r], 95 c o(λ), 104 Sh X , 110 SmX, 108 pI ,54,55 t(R):={t(a),a∈ R},28 Tμ,99 W ,5 Wp,4 0 Tλ,64 Y := Tβ ,80 Λ,89 deg p,99 Θc,99 Ξc, 103 π¯Z,V , 111 λˇ,51 λ t,51 λ \ μ,51 Ad F (R), 140 AF X, 141 S,37 S(c), 99 Fν ,80 Fd, 135 GL(V ) FmX := SmX , 108 P,58 P(μ, s), 79 PA(M,N), 9 SX, 137 Sm(R), 11 SAX, 137 Y,61 d Ac (F ), 97 Ac,d(A), 97 Tr(M), 111 ∇λ(V ),20 πd, 136 σ1 = tr,34

151

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This book gives a unifi ed, complete, and self-contained exposition of the main algebraic theorems of invariant theory for matrices in a characteristic free approach. More precisely, it contains the description of polynomial functions in several variables on the set of m × m matrices with coeffi cients in an infi nite fi eld or even the ring of integers, invariant under simultaneous conjugation. Following Hermann Weyl’s classical approach, the ring of invariants is described by formulating and proving • the fi rst fundamental theorem that describes a set of generators in the ring of invariants, and • the second fundamental theorem that describes relations between these generators. The authors study both the case of matrices over a fi eld of characteristic 0 and the case of matrices over a fi eld of positive characteristic. While the case of characteristic 0 can be treated following a classical approach, the case of positive characteristic (developed by Donkin and Zubkov) is much harder. A presentation of this case requires the develop- ment of a collection of tools. These tools and their application to the study of invariants are exlained in an elementary, self-contained way in the book.

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