Type \(D\) quiver representation varieties, double Grassmannians, and symmetric varieties. (English) Zbl 1475.14093
The study of the representation theory of quivers from a geometric viewpoint is an useful (and well understood) approach to the theory. In this paper, the authors take a step further in their program, that seeks to exhibit the interplay between the (equivariant) geometry of the representation varieties of Dynkin quivers and the geometry of Schubert varieties in multiple flag varieties.
More precisely, let \(Q\) be a type \(D\) quiver and denote the \(\Bbbk\)-representation variety for the dimension vector \(\mathbf d\) by \(\operatorname{rep}_Q(\mathbf d)\), where \(\Bbbk\) is an algebraically closed field; Let \(G=\operatorname{GL}(\mathbf d)\) be the base change group – \(G\) acts on \(\operatorname{rep}_Q(\mathbf d)\). The main result of this work states that there is \(\operatorname{GL}(a+b)\) such that if \(K\) denotes the group \(\operatorname{GL}(a)\times \operatorname{GL}(b)\) embedded diagonally (by blocks) on \(G\), then there exists an order-preserving injective map from the set of \(\operatorname{GL}(\mathbf d)\)-orbit closures in \(\operatorname{rep}_Q(\mathbf d)\) into the set of \(P\)-orbit closures in \(K\backslash G\) for some parabolic subgroup \(P\subset G\). Moreover, any smooth equivalence class of singularities occurring in the domain of this map also occurs in the codomain. The authors also exhibit an homomorphism between the Grothendieck groups of \(K_T(K\backslash G)\to K_{T(\mathbf d)}(\operatorname{rep}_Q(\mathbf d)\)). As a consequence of their work, using general results from the theory of spherical varieties the authors derive a result from [G. Bobiński and G. Zwara, Colloq. Math. 94, No. 2, 285–309 (2002; Zbl 1013.14011)]: the orbit closures in type \(D\) quiver representation are normal, Cohen-Macaulay and have rational singularities.
More precisely, let \(Q\) be a type \(D\) quiver and denote the \(\Bbbk\)-representation variety for the dimension vector \(\mathbf d\) by \(\operatorname{rep}_Q(\mathbf d)\), where \(\Bbbk\) is an algebraically closed field; Let \(G=\operatorname{GL}(\mathbf d)\) be the base change group – \(G\) acts on \(\operatorname{rep}_Q(\mathbf d)\). The main result of this work states that there is \(\operatorname{GL}(a+b)\) such that if \(K\) denotes the group \(\operatorname{GL}(a)\times \operatorname{GL}(b)\) embedded diagonally (by blocks) on \(G\), then there exists an order-preserving injective map from the set of \(\operatorname{GL}(\mathbf d)\)-orbit closures in \(\operatorname{rep}_Q(\mathbf d)\) into the set of \(P\)-orbit closures in \(K\backslash G\) for some parabolic subgroup \(P\subset G\). Moreover, any smooth equivalence class of singularities occurring in the domain of this map also occurs in the codomain. The authors also exhibit an homomorphism between the Grothendieck groups of \(K_T(K\backslash G)\to K_{T(\mathbf d)}(\operatorname{rep}_Q(\mathbf d)\)). As a consequence of their work, using general results from the theory of spherical varieties the authors derive a result from [G. Bobiński and G. Zwara, Colloq. Math. 94, No. 2, 285–309 (2002; Zbl 1013.14011)]: the orbit closures in type \(D\) quiver representation are normal, Cohen-Macaulay and have rational singularities.
Reviewer: Alvaro Rittatore (Montevideo)
MSC:
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 16G20 | Representations of quivers and partially ordered sets |
| 14M27 | Compactifications; symmetric and spherical varieties |
Citations:
Zbl 1013.14011Software:
Macaulay2References:
| [1] | Abeasis, S.; Del Fra, A., Degenerations for the representations of an equioriented quiver of type \(D_m\), Adv. Math., 52, 2, 81-172 (1984) · Zbl 0537.16025 |
| [2] | Allman, J., Grothendieck classes of quiver cycles as iterated residues, Mich. Math. J., 63, 4, 865-888 (2014) · Zbl 1364.14040 |
| [3] | Achinger, P.; Perrin, N., Spherical multiple flags, (Schubert Calculus—Osaka 2012. Schubert Calculus—Osaka 2012, Adv. Stud. Pure Math., vol. 71 (2016), Math. Soc. Japan: Math. Soc. Japan Tokyo), 53-74 · Zbl 1378.14049 |
| [4] | Assem, I.; Simson, D.; Skowroński, A., Elements of the Representation Theory of Associative Algebras. Vol. 1, London Mathematical Society Student Texts, vol. 65 (2006), Cambridge University Press: Cambridge University Press Cambridge, Techniques of representation theory · Zbl 1092.16001 |
| [5] | Auslander, M., Representation theory of finite-dimensional algebras, (Algebraists’ Homage: Papers in Ring Theory and Related Topics. Algebraists’ Homage: Papers in Ring Theory and Related Topics, New Haven, Conn., 1981. Algebraists’ Homage: Papers in Ring Theory and Related Topics. Algebraists’ Homage: Papers in Ring Theory and Related Topics, New Haven, Conn., 1981, Contemp. Math., vol. 13 (1982), Amer. Math. Soc.: Amer. Math. Soc. Providence, R.I.), 27-39 · Zbl 0529.16020 |
| [6] | Buch, A. S.; Fulton, W., Chern class formulas for quiver varieties, Invent. Math., 135, 3, 665-687 (1999) · Zbl 0942.14027 |
| [7] | Buch, A. S.; Fehér, L. M.; Rimányi, R., Positivity of quiver coefficients through Thom polynomials, Adv. Math., 197, 1, 306-320 (2005) · Zbl 1076.14075 |
| [8] | Brion, M.; Kumar, S., Frobenius Splitting Methods in Geometry and Representation Theory, Progress in Mathematics, vol. 231 (2005), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA · Zbl 1072.14066 |
| [9] | Bongartz, K., On degenerations and extensions of finite-dimensional modules, Adv. Math., 121, 2, 245-287 (1996) · Zbl 0862.16007 |
| [10] | Buch, A. S.; Rimányi, R., A formula for non-equioriented quiver orbits of type A, J. Algebraic Geom., 16, 3, 531-546 (2007) · Zbl 1126.14052 |
| [11] | Brion, M., Spherical varieties, (Proceedings of the International Congress of Mathematicians, Vol. 1, 2. Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Zürich, 1994 (1995), Birkhäuser: Birkhäuser Basel), 753-760 · Zbl 0862.14031 |
| [12] | Brion, M., Multiplicity-free subvarieties of flag varieties, (Commutative Algebra. Commutative Algebra, Grenoble/Lyon, 2001. Commutative Algebra. Commutative Algebra, Grenoble/Lyon, 2001, Contemp. Math., vol. 331 (2003), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 13-23 · Zbl 1052.14055 |
| [13] | Brion, M., Spherical varieties, (Highlights in Lie Algebraic Methods. Highlights in Lie Algebraic Methods, Progr. Math., vol. 295 (2012), Birkhäuser/Springer: Birkhäuser/Springer New York), 3-24 · Zbl 1262.14063 |
| [14] | Buch, A. S., Grothendieck classes of quiver varieties, Duke Math. J., 115, 1, 75-103 (2002) · Zbl 1052.14056 |
| [15] | Buch, A. S., Alternating signs of quiver coefficients, J. Am. Math. Soc., 18, 1, 217-237 (2005), (electronic) · Zbl 1061.14050 |
| [16] | Buch, A. S., Quiver coefficients of Dynkin type, Mich. Math. J., 57, 93-120 (2008), Special volume in honor of Melvin Hochster · Zbl 1173.14038 |
| [17] | Bobiński, G.; Zwara, G., Normality of orbit closures for Dynkin quivers of type \(\mathbb{A}_n\), Manuscr. Math., 105, 1, 103-109 (2001) · Zbl 1031.16012 |
| [18] | Bobiński, G.; Zwara, G., Schubert varieties and representations of Dynkin quivers, Colloq. Math., 94, 2, 285-309 (2002) · Zbl 1013.14011 |
| [19] | Cartan, E., Sur une classe remarquable d’espaces de Riemann, Bull. Soc. Math. Fr., 54, 214-264 (1926) · JFM 52.0425.01 |
| [20] | Cartan, E., Sur une classe remarquable d’espaces de Riemann. II, Bull. Soc. Math. Fr., 55, 114-134 (1927) |
| [21] | Chriss, N.; Ginzburg, V., Representation Theory and Complex Geometry, Modern Birkhäuser Classics (2010), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA, Reprint of the 1997 edition · Zbl 1185.22001 |
| [22] | Derksen, H.; Weyman, J., An Introduction to Quiver Representations, Graduate Studies in Mathematics, vol. 184 (2017), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1426.16001 |
| [23] | Fehér, L. M.; Patakfalvi, Z., The incidence class and the hierarchy of orbits, Cent. Eur. J. Math., 7, 3, 429-441 (2009) · Zbl 1184.32008 |
| [24] | Fehér, L.; Rimányi, R., Classes of degeneracy loci for quivers: the Thom polynomial point of view, Duke Math. J., 114, 2, 193-213 (2002) · Zbl 1054.14010 |
| [25] | Walter, R.; Santos, F.; Rittatore, A., Actions and Invariants of Algebraic Groups, Monographs and Research Notes in Mathematics (2017), CRC Press: CRC Press Boca Raton, FL · Zbl 1354.14069 |
| [26] | Grayson, D. R.; Stillman, M. E., Macaulay2, a software system for research in algebraic geometry, Available at |
| [27] | Hesselink, W., Singularities in the nilpotent scheme of a classical group, Trans. Am. Math. Soc., 222, 1-32 (1976) · Zbl 0332.14017 |
| [28] | Kinser, R., K-polynomials of type a quiver orbit closures and lacing diagrams, (Representations of Algebras. Representations of Algebras, Contemp. Math., vol. 705 (2018), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 99-114 · Zbl 1398.14052 |
| [29] | Kinser, R.; Knutson, A.; Rajchgot, J., Three combinatorial formulas for type A quiver polynomials and K-polynomials, Duke Math. J., 168, 4, 505-551 (2019) · Zbl 1480.16035 |
| [30] | Kazhdan, D.; Lusztig, G., Representations of Coxeter groups and Hecke algebras, Invent. Math., 53, 2, 165-184 (1979) · Zbl 0499.20035 |
| [31] | Knutson, A.; Lam, T.; Speyer, D. E., Projections of Richardson varieties, J. Reine Angew. Math., 687, 133-157 (2014) · Zbl 1345.14047 |
| [32] | Knutson, A.; Miller, E.; Shimozono, M., Four positive formulae for type A quiver polynomials, Invent. Math., 166, 2, 229-325 (2006) · Zbl 1107.14046 |
| [33] | Kinser, R.; Rajchgot, J., Type A quiver loci and Schubert varieties, J. Commut. Algebra, 7, 2, 265-301 (2015) · Zbl 1351.14031 |
| [34] | Knutson, A.; Woo, A.; Yong, A., Singularities of Richardson varieties, Math. Res. Lett., 20, 2, 391-400 (2013) · Zbl 1298.14053 |
| [35] | Lakshmibai, V.; Magyar, P., Degeneracy schemes, quiver schemes, and Schubert varieties, Int. Math. Res. Not., 12, 627-640 (1998) · Zbl 0936.14001 |
| [36] | Matsumura, H., Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, vol. 8 (1989), Cambridge University Press: Cambridge University Press Cambridge, Translated from the Japanese by M. Reid · Zbl 0666.13002 |
| [37] | Merkurjev, A. S., Equivariant K-theory, (Handbook of K-Theory. Vol. 1, 2 (2005), Springer: Springer Berlin), 925-954 · Zbl 1108.19002 |
| [38] | Mumford, D.; Fogarty, J.; Kirwan, F., Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34 (1994), Springer-Verlag: Springer-Verlag Berlin · Zbl 0797.14004 |
| [39] | Miller, E., Alternating formulas for K-theoretic quiver polynomials, Duke Math. J., 128, 1, 1-17 (2005) · Zbl 1099.05079 |
| [40] | Matsuki, T.; Ōshima, T., Embeddings of discrete series into principal series, (The Orbit Method in Representation Theory. The Orbit Method in Representation Theory, Copenhagen, 1988. The Orbit Method in Representation Theory. The Orbit Method in Representation Theory, Copenhagen, 1988, Progr. Math., vol. 82 (1990), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 147-175 · Zbl 0746.22011 |
| [41] | Mehta, V. B.; Ramanathan, A., Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. Math. (2), 122, 1, 27-40 (1985) · Zbl 0601.14043 |
| [42] | Musili, C.; Seshadri, C. S., Schubert varieties and the variety of complexes, (Arithmetic and Geometry, Vol. II. Arithmetic and Geometry, Vol. II, Progr. Math., vol. 36 (1983), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 329-359 · Zbl 0567.14030 |
| [43] | Mars, J. G.M.; Springer, T. A., Hecke algebra representations related to spherical varieties, Represent. Theory, 2, 33-69 (1998) · Zbl 0887.14026 |
| [44] | Magyar, P.; Weyman, J.; Zelevinsky, A., Multiple flag varieties of finite type, Adv. Math., 141, 1, 97-118 (1999) · Zbl 0951.14034 |
| [45] | Perrin, N., On the geometry of spherical varieties, Transform. Groups, 19, 1, 171-223 (2014) · Zbl 1309.14001 |
| [46] | Rimányi, R., Quiver polynomials in iterated residue form, J. Algebraic Comb., 40, 2, 527-542 (2014) · Zbl 1348.14065 |
| [47] | Rimányi, R., Motivic characteristic classes in cohomological Hall algebras, Adv. Math., 360, Article 106888 pp. (2020) · Zbl 1439.14033 |
| [48] | Riedtmann, C.; Zwara, G., Orbit closures and rank schemes, Comment. Math. Helv., 88, 1, 55-84 (2013) · Zbl 1266.14038 |
| [49] | Schiffler, R., Quiver Representations, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. (2014), Springer: Springer Cham · Zbl 1310.16015 |
| [50] | Smirnov, E., Orbites d’un sous-groupe de Borel dans le produit de deux grassmanniennes (2007), Institut Fourier, Université Grenoble I, PhD thesis |
| [51] | Smirnov, E. Yu., Resolutions of singularities for Schubert varieties in double Grassmannians, Funkc. Anal. Prilozh., 42, 2, 56-67 (2008), 96 · Zbl 1206.14074 |
| [52] | Smirnov, E., Multiple flag varieties, (Proceedings of the Seminar on Algebra and Geometry of the Samara University. Proceedings of the Seminar on Algebra and Geometry of the Samara University, Moscow. Proceedings of the Seminar on Algebra and Geometry of the Samara University. Proceedings of the Seminar on Algebra and Geometry of the Samara University, Moscow, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., vol. 147 (2018)), 84-119 |
| [53] | Springer, T. A., Some results on algebraic groups with involutions, (Algebraic Groups and Related Topics. Algebraic Groups and Related Topics, Kyoto/Nagoya, 1983. Algebraic Groups and Related Topics. Algebraic Groups and Related Topics, Kyoto/Nagoya, 1983, Adv. Stud. Pure Math., vol. 6 (1985), North-Holland: North-Holland Amsterdam), 525-543 · Zbl 1146.14026 |
| [54] | Springer, T. A., Linear Algebraic Groups, Modern Birkhäuser Classics (2009), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA · Zbl 1202.20048 |
| [55] | The Stacks Project Authors, Stacks project (2018) |
| [56] | Smith, K. E.; Zhang, W., Frobenius splitting in commutative algebra, (Commutative Algebra and Noncommutative Algebraic Geometry. Vol. I. Commutative Algebra and Noncommutative Algebraic Geometry. Vol. I, Math. Sci. Res. Inst. Publ., vol. 67 (2015), Cambridge Univ. Press: Cambridge Univ. Press New York), 291-345 · Zbl 1359.13006 |
| [57] | Thomason, R. W., Algebraic K-theory of group scheme actions, (Algebraic Topology and Algebraic K-Theory. Algebraic Topology and Algebraic K-Theory, Princeton, N.J., 1983. Algebraic Topology and Algebraic K-Theory. Algebraic Topology and Algebraic K-Theory, Princeton, N.J., 1983, Ann. of Math. Stud., vol. 113 (1987), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ), 539-563 · Zbl 0701.19002 |
| [58] | Thomason, R. W., Equivariant algebraic vs. topological K-homology Atiyah-Segal-style, Duke Math. J., 56, 3, 589-636 (1988) · Zbl 0655.55002 |
| [59] | Woo, A.; Wyser, B.; Yong, A., Governing singularities of symmetric orbit closures, Algebra Number Theory, 12, 1, 173-225 (2018) · Zbl 1427.14106 |
| [60] | Wyser, B. J., The Bruhat order on clans, J. Algebraic Comb., 44, 3, 495-517 (2016) · Zbl 1350.05182 |
| [61] | Yamamoto, A., Orbits in the flag variety and images of the moment map for classical groups. I, Represent. Theory, 1, 329-404 (1997) · Zbl 0887.22017 |
| [62] | Yong, A., On combinatorics of quiver component formulas, J. Algebraic Comb., 21, 3, 351-371 (2005) · Zbl 1078.05087 |
| [63] | Zelevinsky, A. V., Two remarks on graded nilpotent classes, Usp. Mat. Nauk, 40, 1(241), 199-200 (1985) · Zbl 0577.20032 |
| [64] | Zwara, G., Degenerations for modules over representation-finite algebras, Proc. Am. Math. Soc., 127, 5, 1313-1322 (1999) · Zbl 0927.16008 |
| [65] | Zwara, G., Smooth morphisms of module schemes, Proc. Lond. Math. Soc. (3), 84, 3, 539-558 (2002) · Zbl 1054.16009 |
| [66] | Zwara, G., Singularities of orbit closures in module varieties, (Representations of Algebras and Related Topics. Representations of Algebras and Related Topics, EMS Ser. Congr. Rep. (2011), Eur. Math. Soc.: Eur. Math. Soc. Zürich), 661-725 · Zbl 1307.14069 |
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