K-polynomials of type A quiver orbit closures and lacing diagrams. (English) Zbl 1398.14052
Leuschke, Graham J. (ed.) et al., Representations of algebras. 17th workshop and international conference on representations of algebras (ICRA 2016), Syracuse University, Syracuse, NY, USA, August 10–19, 2016. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-3576-9/pbk; 978-1-4704-4721-2/ebook). Contemporary Mathematics 705, 99-114 (2018).
Summary: This article contains an overview of the author’s joint work with A. Knutson and J. Rajchgot [“Three combinatorial formulas for type \(A\) quiver polynomials and \(K\)-polynomials”, Preprint, arXiv:1503.05880] on \(K\)-polynomials of orbit closures for type \(A\) quivers. It is written to an audience interested in interactions between representations of algebras, algebraic geometry, and commutative algebra. A few open problems resulting from the work are also explained.
For the entire collection see [Zbl 1390.16001].
For the entire collection see [Zbl 1390.16001].
MSC:
| 14M12 | Determinantal varieties |
| 05E15 | Combinatorial aspects of groups and algebras (MSC2010) |
| 14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |
| 19E08 | \(K\)-theory of schemes |
| 16G20 | Representations of quivers and partially ordered sets |
References:
| [1] | Abeasis, S.; Del Fra, A., Degenerations for the representations of a quiver of type \({\mathcal{A}}_m\), J. Algebra, 93, 2, 376-412 (1985) · Zbl 0598.16030 · doi:10.1016/0021-8693(85)90166-8 |
| [2] | Allman, Justin, Grothendieck classes of quiver cycles as iterated residues, Michigan Math. J., 63, 4, 865-888 (2014) · Zbl 1364.14040 · doi:10.1307/mmj/1417799229 |
| [3] | Auslander, Maurice; Reiten, Idun; Smal\o, Sverre O., Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics 36, xiv+425 pp. (1997), Cambridge University Press, Cambridge · Zbl 0834.16001 |
| [4] | Assem, Ibrahim; Simson, Daniel; Skowro\'nski, Andrzej, Elements of the representation theory of associative algebras. Vol. 1, London Mathematical Society Student Texts 65, x+458 pp. (2006), Cambridge University Press, Cambridge · Zbl 1092.16001 · doi:10.1017/CBO9780511614309 |
| [5] | Buch, Anders Skovsted; Fulton, William, Chern class formulas for quiver varieties, Invent. Math., 135, 3, 665-687 (1999) · Zbl 0942.14027 · doi:10.1007/s002220050297 |
| [6] | Buch, Anders S.; Feh\'er, L\'aszl\'o M.; Rim\'anyi, Rich\'ard, Positivity of quiver coefficients through Thom polynomials, Adv. Math., 197, 1, 306-320 (2005) · Zbl 1076.14075 · doi:10.1016/j.aim.2004.10.019 |
| [7] | Bongartz, Klaus, Degenerations for representations of tame quivers, Ann. Sci. \'Ecole Norm. Sup. (4), 28, 5, 647-668 (1995) · Zbl 0844.16007 |
| [8] | Bongartz, Klaus, On degenerations and extensions of finite-dimensional modules, Adv. Math., 121, 2, 245-287 (1996) · Zbl 0862.16007 · doi:10.1006/aima.1996.0053 |
| [9] | Bongartz, Klaus, Some geometric aspects of representation theory. Algebras and modules, I, Trondheim, 1996, CMS Conf. Proc. 23, 1-27 (1998), Amer. Math. Soc., Providence, RI · Zbl 0915.16008 |
| [10] | Buch, Anders S.; Rim\'anyi, Rich\'ard, Specializations of Grothendieck polynomials, C. R. Math. Acad. Sci. Paris, 339, 1, 1-4 (2004) · Zbl 1051.14062 · doi:10.1016/j.crma.2004.04.015 |
| [11] | Buch, Anders Skovsted; Rim\'anyi, Rich\'ard, A formula for non-equioriented quiver orbits of type \(A\), J. Algebraic Geom., 16, 3, 531-546 (2007) · Zbl 1126.14052 · doi:10.1090/S1056-3911-07-00441-9 |
| [12] | Brion, Michel, Multiplicity-free subvarieties of flag varieties. Commutative algebra, Grenoble/Lyon, 2001, Contemp. Math. 331, 13-23 (2003), Amer. Math. Soc., Providence, RI · Zbl 1052.14055 · doi:10.1090/conm/331/05900 |
| [13] | Buch, Anders Skovsted, Grothendieck classes of quiver varieties, Duke Math. J., 115, 1, 75-103 (2002) · Zbl 1052.14056 · doi:10.1215/S0012-7094-02-11513-0 |
| [14] | Buch, Anders Skovsted, A Littlewood-Richardson rule for the \(K\)-theory of Grassmannians, Acta Math., 189, 1, 37-78 (2002) · Zbl 1090.14015 · doi:10.1007/BF02392644 |
| [15] | Buch, Anders Skovsted, Alternating signs of quiver coefficients, J. Amer. Math. Soc., 18, 1, 217-237 (2005) · Zbl 1061.14050 · doi:10.1090/S0894-0347-04-00473-4 |
| [16] | Buch, Anders Skovsted, Combinatorial \(K\)-theory. Topics in cohomological studies of algebraic varieties, Trends Math., 87-103 (2005), Birkh\`“auser, Basel · Zbl 1052.14056 · doi:10.1007/3-7643-7342-3\_3 |
| [17] | Buch, Anders Skovsted, Quiver coefficients of Dynkin type, Michigan Math. J., 57, 93-120 (2008) · Zbl 1173.14038 · doi:10.1307/mmj/1220879399 |
| [18] | Bobi\'nski, Grzegorz; Zwara, Grzegorz, Normality of orbit closures for Dynkin quivers of type \(\mathbb{A}_n\), Manuscripta Math., 105, 1, 103-109 (2001) · Zbl 1031.16012 · doi:10.1007/PL00005871 |
| [19] | Bobi\'nski, Grzegorz; Zwara, Grzegorz, Schubert varieties and representations of Dynkin quivers, Colloq. Math., 94, 2, 285-309 (2002) · Zbl 1013.14011 · doi:10.4064/cm94-2-10 |
| [20] | Fomin, Sergey; Kirillov, Anatol N., Grothendieck polynomials and the Yang-Baxter equation. Formal power series and algebraic combinatorics/S\'eries formelles et combinatoire alg\'ebrique, 183-189 (sd), DIMACS, Piscataway, NJ |
| [21] | Fulton, William; Lascoux, Alain, A Pieri formula in the Grothendieck ring of a flag bundle, Duke Math. J., 76, 3, 711-729 (1994) · Zbl 0840.14007 · doi:10.1215/S0012-7094-94-07627-8 |
| [22] | Feh\'er, L\'aszl\'o; Rim\'anyi, Rich\'ard, Classes of degeneracy loci for quivers: the Thom polynomial point of view, Duke Math. J., 114, 2, 193-213 (2002) · Zbl 1054.14010 · doi:10.1215/S0012-7094-02-11421-5 |
| [23] | Feh\'er, L\'aszl\'o M.; Rim\'anyi, Rich\'ard, Calculation of Thom polynomials and other cohomological obstructions for group actions. Real and complex singularities, Contemp. Math. 354, 69-93 (2004), Amer. Math. Soc., Providence, RI · Zbl 1074.32008 · doi:10.1090/conm/354/06475 |
| [24] | Fulton, William, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J., 65, 3, 381-420 (1992) · Zbl 0788.14044 · doi:10.1215/S0012-7094-92-06516-1 |
| [25] | Fulton, William, Universal Schubert polynomials, Duke Math. J., 96, 3, 575-594 (1999) · Zbl 0981.14022 · doi:10.1215/S0012-7094-99-09618-7 |
| [26] | Gei\ss, Christof; Leclerc, Bernard; Schr\`“oer, Jan, Quivers with relations for symmetrizable Cartan matrices III: Convolution algebras, Represent. Theory, 20, 375-413 (2016) · Zbl 1362.16018 · doi:10.1090/ert/487 |
| [27] | Huisgen-Zimmermann, B., Fine and coarse moduli spaces in the representation theory of finite dimensional algebras. Expository lectures on representation theory, Contemp. Math. 607, 1-34 (2014), Amer. Math. Soc., Providence, RI · Zbl 1327.16003 · doi:10.1090/conm/607/12086 |
| [28] | Kinser, Ryan, Rank functions on rooted tree quivers, Duke Math. J., 152, 1, 27-92 (2010) · Zbl 1237.16011 · doi:10.1215/00127094-2010-006 |
| [29] | Kinser, Ryan, Tree modules and counting polynomials, Algebr. Represent. Theory, 16, 5, 1333-1347 (2013) · Zbl 1306.16007 · doi:10.1007/s10468-012-9359-x |
| [30] | Ryan Kinser, Allen Knutson, and Jenna Rajchgot, Three combinatorial formulas for type \(A\) quiver polynomials and \(K\)-polynomials, \href{http://arXiv.org/abs/1503.05880}arxiv:1503.05880. · Zbl 1480.16035 |
| [31] | Knutson, Allen; Miller, Ezra, Subword complexes in Coxeter groups, Adv. Math., 184, 1, 161-176 (2004) · Zbl 1069.20026 · doi:10.1016/S0001-8708(03)00142-7 |
| [32] | Knutson, Allen; Miller, Ezra, Gr\`“obner geometry of Schubert polynomials, Ann. of Math. (2), 161, 3, 1245-1318 (2005) · Zbl 1089.14007 · doi:10.4007/annals.2005.161.1245 |
| [33] | Knutson, Allen; Miller, Ezra; Shimozono, Mark, Four positive formulae for type \(A\) quiver polynomials, Invent. Math., 166, 2, 229-325 (2006) · Zbl 1107.14046 · doi:10.1007/s00222-006-0505-0 |
| [34] | Allen Knutson, Frobenius splitting and Mobius inversion, \href{http://arxiv.org/abs/0902.1930v1}arxiv:0902.1930v1. |
| [35] | Kinser, Ryan; Rajchgot, Jenna, Type \(A\) quiver loci and Schubert varieties, J. Commut. Algebra, 7, 2, 265-301 (2015) · Zbl 1351.14031 · doi:10.1216/JCA-2015-7-2-265 |
| [36] | Krause, Henning, Maps between tree and band modules, J. Algebra, 137, 1, 186-194 (1991) · Zbl 0715.16007 · doi:10.1016/0021-8693(91)90088-P |
| [37] | Lenart, Cristian, Noncommutative Schubert calculus and Grothendieck polynomials, Adv. Math., 143, 1, 159-183 (1999) · Zbl 0978.05074 · doi:10.1006/aima.1998.1795 |
| [38] | Lenart, Cristian, Combinatorial aspects of the \(K\)-theory of Grassmannians, Ann. Comb., 4, 1, 67-82 (2000) · Zbl 0958.05128 · doi:10.1007/PL00001276 |
| [39] | Lakshmibai, V.; Magyar, Peter, Degeneracy schemes, quiver schemes, and Schubert varieties, Internat. Math. Res. Notices, 12, 627-640 (1998) · Zbl 0936.14001 · doi:10.1155/S1073792898000397 |
| [40] | L\H orincz, Andr\'as Cristian, The \(b\)-functions of semi-invariants of quivers, J. Algebra, 482, 346-363 (2017) · Zbl 1411.16015 · doi:10.1016/j.jalgebra.2017.03.028 |
| [41] | \bysame, Singularities of zero sets of semi-invariants for quivers, \href{http://arXiv.org/abs/1509.04170}arxiv:1509.04170. |
| [42] | Lascoux, Alain; Sch\`“utzenberger, Marcel-Paul, Structure de Hopf de l”anneau de cohomologie et de l’anneau de Grothendieck d’une vari\'et\'e de drapeaux, C. R. Acad. Sci. Paris S\'er. I Math., 295, 11, 629-633 (1982) · Zbl 0542.14030 |
| [43] | Lusztig, G., Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc., 3, 2, 447-498 (1990) · Zbl 0703.17008 · doi:10.2307/1990961 |
| [44] | Lusztig, G., Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc., 4, 2, 365-421 (1991) · Zbl 0738.17011 · doi:10.2307/2939279 |
| [45] | Miller, Ezra; Sturmfels, Bernd, Combinatorial commutative algebra, Graduate Texts in Mathematics 227, xiv+417 pp. (2005), Springer-Verlag, New York · Zbl 1090.13001 |
| [46] | Perrin, Nicolas, On the geometry of spherical varieties, Transform. Groups, 19, 1, 171-223 (2014) · Zbl 1309.14001 · doi:10.1007/s00031-014-9254-0 |
| [47] | Riedtmann, Christine, Degenerations for representations of quivers with relations, Ann. Sci. \'Ecole Norm. Sup. (4), 19, 2, 275-301 (1986) · Zbl 0603.16025 |
| [48] | Richard Rimanyi, On the cohomological Hall algebra of Dynkin quivers, \href{http://arxiv.org/abs/1303.3399}arxiv:1303.3399. |
| [49] | Rim\'anyi, R., Quiver polynomials in iterated residue form, J. Algebraic Combin., 40, 2, 527-542 (2014) · Zbl 1348.14065 · doi:10.1007/s10801-013-0497-y |
| [50] | Ringel, Claus Michael, The rational invariants of the tame quivers, Invent. Math., 58, 3, 217-239 (1980) · Zbl 0433.15009 · doi:10.1007/BF01390253 |
| [51] | Ringel, Claus Michael, Hall algebras and quantum groups, Invent. Math., 101, 3, 583-591 (1990) · Zbl 0735.16009 · doi:10.1007/BF01231516 |
| [52] | Ringel, Claus Michael, Exceptional modules are tree modules, Proceedings of the Sixth Conference of the International Linear Algebra Society (Chemnitz, 1996). Linear Algebra Appl., 275/276, 471-493 (1998) · Zbl 0964.16014 · doi:10.1016/S0024-3795(97)10046-5 |
| [53] | Ringel, Claus Michael, Distinguished bases of exceptional modules. Algebras, quivers and representations, Abel Symp. 8, 253-274 (2013), Springer, Heidelberg · Zbl 1319.16015 · doi:10.1007/978-3-642-39485-0\_11 |
| [54] | Ringel, Claus Michael, Indecomposable representations of the Kronecker quivers, Proc. Amer. Math. Soc., 141, 1, 115-121 (2013) · Zbl 1284.16017 · doi:10.1090/S0002-9939-2012-11296-1 |
| [55] | Riedtmann, Christine; Zwara, Grzegorz, Orbit closures and rank schemes, Comment. Math. Helv., 88, 1, 55-84 (2013) · Zbl 1266.14038 · doi:10.4171/CMH/278 |
| [56] | Schofield, A. H., Representation of rings over skew fields, London Mathematical Society Lecture Note Series 92, xii+223 pp. (1985), Cambridge University Press, Cambridge · Zbl 0571.16001 · doi:10.1017/CBO9780511661914 |
| [57] | Schiffler, Ralf, Quiver representations, CMS Books in Mathematics/Ouvrages de Math\'ematiques de la SMC, xii+230 pp. (2014), Springer, Cham · Zbl 1310.16015 · doi:10.1007/978-3-319-09204-1 |
| [58] | Weist, Thorsten, Tree modules of the generalised Kronecker quiver, J. Algebra, 323, 4, 1107-1138 (2010) · Zbl 1219.16018 · doi:10.1016/j.jalgebra.2009.11.033 |
| [59] | Weist, Thorsten, Tree modules, Bull. Lond. Math. Soc., 44, 5, 882-898 (2012) · Zbl 1279.16015 · doi:10.1112/blms/bds019 |
| [60] | Woo, Alexander; Yong, Alexander, A Gr\`“obner basis for Kazhdan-Lusztig ideals, Amer. J. Math., 134, 4, 1089-1137 (2012) · Zbl 1262.13044 · doi:10.1353/ajm.2012.0031 |
| [61] | Zelevinski\u\i, A. V., Two remarks on graded nilpotent classes, Uspekhi Mat. Nauk, 40, 1(241), 199-200 (1985) · Zbl 0577.20032 |
| [62] | Zwara, Grzegorz, Degenerations for representations of extended Dynkin quivers, Comment. Math. Helv., 73, 1, 71-88 (1998) · Zbl 0902.16016 · doi:10.1007/s000140050046 |
| [63] | Zwara, Grzegorz, Degenerations for modules over representation-finite algebras, Proc. Amer. Math. Soc., 127, 5, 1313-1322 (1999) · Zbl 0927.16008 · doi:10.1090/S0002-9939-99-04714-0 |
| [64] | Zwara, Grzegorz, Degenerations of finite-dimensional modules are given by extensions, Compositio Math., 121, 2, 205-218 (2000) · Zbl 0957.16007 · doi:10.1023/A:1001778532124 |
| [65] | Zwara, G., Smooth morphisms of module schemes, Proc. London Math. Soc. (3), 84, 3, 539-558 (2002) · Zbl 1054.16009 · doi:10.1112/S0024611502013382 |
| [66] | Zwara, Grzegorz, Singularities of orbit closures in module varieties. Representations of algebras and related topics, EMS Ser. Congr. Rep., 661-725 (2011), Eur. Math. Soc., Z\`“urich · Zbl 1307.14069 · doi:10.4171/101-1/13 |
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