Content systems and deformations of cyclotomic KLR algebras of type $A$ and $C$

Evseev, Anton; Mathas, Andrew

doi:10.5802/art.8

Content systems and deformations of cyclotomic KLR algebras of type

A

and

C

Evseev, Anton ¹; Mathas, Andrew ²

¹ School of Mathematics, University of Birmingham Edgbaston B15 2TT United Kingdom
² School of Mathematics and Statistics, University of Sydney Carslaw F07 Sydney NSW 2006 Australia

Annals of Representation Theory, Volume 1 (2024) no. 2, pp. 193-297.

Abstract

This paper initiates a systematic study of the cyclotomic KLR algebras of affine types $A$ and $C$ . We start by introducing a graded deformation of these algebras and then constructing all of the irreducible representations of the deformed cyclotomic KLR algebras using content systems, which are recursively defined using Rouquier’s $Q$ -polynomials. This leads to a generalisation of the Young’s seminormal forms for the symmetric groups in the KLR setting. Quite amazingly, the same theory captures the representation theory of the cyclotomic KLR algebras of affine types $A$ and $C$ , with the main difference being that the definition of the residue sequence of a tableau depends on the Cartan type. We use our semisimple deformations to construct two “dual” cellular bases for the non-semisimple KLR algebras of affine types $A$ and $C$ . As applications we recover many of the main features from the representation theory in type $A$ , simultaneously proving them for the cyclotomic KLR algebras of types $A$ and $C$ . These results are completely new in type $C$ and we, usually, give more direct proofs in type $A$ . In particular, we show that these algebras categorify the irreducible integrable highest weight modules of the corresponding Kac–Moody algebras, we construct and classify their simple modules, we investigate links with canonical bases and we generalise Kleshchev’s modular branching rules to these algebras.

Received: 2023-08-26
Revised: 2023-12-20
Accepted: 2024-01-31
Published online: 2024-05-31

DOI: 10.5802/art.8

Classification: 20C08, 18N25, 20G44, 05E10
Keywords: Cyclotomic KLR algebras, quiver Hecke algebras, categorification, quantum groups, representation theory, cellular algebras, Specht modules, seminormal forms

Author's affiliations:

Evseev, Anton ¹; Mathas, Andrew ²

¹ School of Mathematics, University of Birmingham Edgbaston B15 2TT United Kingdom
² School of Mathematics and Statistics, University of Sydney Carslaw F07 Sydney NSW 2006 Australia

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Evseev, Anton; Mathas, Andrew. Content systems and deformations of cyclotomic KLR algebras of type $A$ and $C$. Annals of Representation Theory, Volume 1 (2024) no. 2, pp. 193-297. doi : 10.5802/art.8. https://art.centre-mersenne.org/articles/10.5802/art.8/

References
Cited by

[1] Ariki, Susumu On the decomposition numbers of the Hecke algebra of $G (m, 1, n)$ , J. Math. Kyoto Univ., Volume 36 (1996) no. 4, pp. 789-808 | MR | Zbl

[2] Ariki, Susumu On the classification of simple modules for cyclotomic Hecke algebras of type $G (m, 1, n)$ and Kleshchev multipartitions, Osaka J. Math., Volume 38 (2001) no. 4, pp. 827-837 | MR | Zbl

[3] Ariki, Susumu Representations of quantum algebras and combinatorics of Young tableaux, University Lecture Series, 26, American Mathematical Society, 2002 (translated from the 2000 Japanese edition and revised by the author) | Zbl

[4] Ariki, Susumu Proof of the modular branching rule for cyclotomic Hecke algebras, J. Algebra, Volume 306 (2006) no. 1, pp. 290-300 | DOI | MR | Zbl

[5] Ariki, Susumu; Park, Euiyong Representation type of finite quiver Hecke algebras of type $A_{2 ℓ}^{(2)}$ , J. Algebra, Volume 397 (2014), pp. 457-488 | DOI | MR | Zbl

[6] Ariki, Susumu; Park, Euiyong Representation type of finite quiver Hecke algebras of type $C_{l}^{(1)}$ , Osaka J. Math., Volume 53 (2016) no. 2, pp. 463-488 | Zbl

[7] Ariki, Susumu; Park, Euiyong Representation type of finite quiver Hecke algebras of type $D_{ℓ + 1}^{(2)}$ , Trans. Am. Math. Soc., Volume 368 (2016) no. 5, pp. 3211-3242 | DOI | MR | Zbl

[8] Ariki, Susumu; Park, Euiyong; Speyer, Liron Specht modules for quiver Hecke algebras of type $C$ , Publ. Res. Inst. Math. Sci., Ser. A, Volume 55 (2019) no. 3, pp. 565-626 | DOI | MR | Zbl

[9] Bowman, Christopher The many integral graded cellular bases of Hecke algebras of complex reflection groups, Am. J. Math., Volume 144 (2022) no. 2, pp. 437-504 | DOI | MR | Zbl

[10] Brundan, Jonathan; Kleshchev, Alexander Blocks of cyclotomic Hecke algebras and Khovanov–Lauda algebras, Invent. Math., Volume 178 (2009) no. 3, pp. 451-484 | DOI | MR | Zbl

[11] Brundan, Jonathan; Kleshchev, Alexander Graded decomposition numbers for cyclotomic Hecke algebras, Adv. Math., Volume 222 (2009) no. 6, pp. 1883-1942 | DOI | MR | Zbl

[12] Brundan, Jonathan; Kleshchev, Alexander The degenerate analogue of Ariki’s categorification theorem, Math. Z., Volume 266 (2010) no. 4, pp. 877-919 | DOI | MR | Zbl

[13] Brundan, Jonathan; Kleshchev, Alexander; Wang, Weiqiang Graded Specht modules, J. Reine Angew. Math., Volume 655 (2011), pp. 61-87 | DOI | MR | Zbl

[14] Brundan, Jonathan; Stroppel, Catharina Highest weight categories arising from Khovanov’s diagram algebra. II. Koszulity, Transform. Groups, Volume 15 (2010) no. 1, pp. 1-45 | DOI | MR | Zbl

[15] Chlouveraki, Maria; Jacon, Nicolas Schur elements for the Ariki–Koike algebra and applications, J. Algebr. Comb., Volume 35 (2012) no. 2, pp. 291-311 | DOI | MR | Zbl

[16] Chuang, Joseph; Rouquier, Raphaël Derived equivalences for symmetric groups and ${𝔰𝔩}_{2}$ -categorification, Ann. Math., Volume 167 (2008) no. 1, pp. 245-298 | DOI | MR | Zbl

[17] Chung, Christopher; Mathas, Andrew; Speyer, Liron Some graded decomposition matrices of finite quiver Hecke algebras in type C (2023), p. in preparation

[18] Curtis, Charles W.; Reiner, Irving Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, 11, Interscience Publishers, 1962 | Zbl

[19] Dipper, Richard; James, Gordon D.; Mathas, Andrew Cyclotomic $q$ -Schur algebras, Math. Z., Volume 229 (1998) no. 3, pp. 385-416 | DOI | MR | Zbl

[20] Evseev, Anton On graded decomposition numbers for cyclotomic Hecke algebras in quantum characteristic 2, Bull. Lond. Math. Soc., Volume 46 (2014) no. 4, pp. 725-731 | DOI | MR | Zbl

[21] Graham, John J.; Lehrer, Gustav I. Cellular algebras, Invent. Math., Volume 123 (1996) no. 1, pp. 1-34 | DOI | MR | Zbl

[22] Grojnowski, Ian Affine ${\hat{s l}}_{p}$ controls the modular representation theory of the symmetric group and related algebras, 1999, p. preprint

[23] Hayashi, Takahiro $q$ -Analogues of Clifford and Weyl algebras—spinor and oscillator representations of quantum enveloping algebras, Commun. Math. Phys., Volume 127 (1990) no. 1, pp. 129-144 | DOI | MR | Zbl

[24] Hu, Jun; Mathas, Andrew Graded cellular bases for the cyclotomic Khovanov–Lauda–Rouquier algebras of type $A$ , Adv. Math., Volume 225 (2010) no. 2, pp. 598-642 | DOI | MR | Zbl

[25] Hu, Jun; Mathas, Andrew Graded induction for Specht modules, Int. Math. Res. Not., Volume 2012 (2012) no. 6, pp. 1230-1263 | DOI | MR | Zbl

[26] Hu, Jun; Mathas, Andrew Seminormal forms and cyclotomic quiver Hecke algebras of type A, Math. Ann., Volume 364 (2016) no. 3-4, pp. 1189-1254 | DOI | MR | Zbl

[27] Hu, Jun; Mathas, Andrew Fayers’ conjecture and the socles of cyclotomic Weyl modules, Trans. Am. Math. Soc., Volume 371 (2019) no. 2, pp. 1271-1307 | DOI | MR | Zbl

[28] Hu, Jun; Shi, Lei Graded dimensions and monomial bases for the cyclotomic quiver Hecke algebras (2024) (to appear in Communications in Contemporary Mathematics) | arXiv | DOI

[29] James, Gordon D. The representation theory of the symmetric groups, Lecture Notes in Mathematics, 682, Springer, 1978 | DOI | Zbl

[30] Kac, Victor G. Infinite-dimensional Lie algebras, Cambridge University Press, 1990 | DOI | Zbl

[31] Kang, Seok-Jin; Kashiwara, Masaki Categorification of highest weight modules via Khovanov–Lauda–Rouquier algebras, Invent. Math., Volume 190 (2012) no. 3, pp. 699-742 | DOI | MR | Zbl

[32] Kang, Seok-Jin; Kashiwara, Masaki; Kim, Myungho Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras, Invent. Math., Volume 211 (2018) no. 2, pp. 591-685 | DOI | MR | Zbl

[33] Kashiwara, Masaki On crystal bases of the $Q$ -analogue of universal enveloping algebras, Duke Math. J., Volume 63 (1991) no. 2, pp. 465-516 | DOI | MR | Zbl

[34] Kashiwara, Masaki On crystal bases, Representations of groups (Banff, AB, 1994) (CMS Conference Proceedings), Volume 16, American Mathematical Society, 1995, pp. 155-197 | MR | Zbl

[35] Kashiwara, Masaki Biadjointness in cyclotomic Khovanov–Lauda–Rouquier algebras, Publ. Res. Inst. Math. Sci., Ser. A, Volume 48 (2012) no. 3, pp. 501-524 | DOI | MR | Zbl

[36] Khovanov, Mikhail; Lauda, Aaron D. A diagrammatic approach to categorification of quantum groups. I, Represent. Theory, Volume 13 (2009), pp. 309-347 | DOI | MR | Zbl

[37] Kim, Jeong-Ah; Shin, Dong-Uy Crystal bases and generalized Lascoux–Leclerc–Thibon (LLT) algorithm for the quantum affine algebra $U_{q} (C_{n}^{(1)})$ , J. Math. Phys., Volume 45 (2004) no. 12, pp. 4878-4895 | DOI | MR | Zbl

[38] Kleshchev, Alexander Branching rules for modular representations of symmetric groups. III. Some corollaries and a problem of Mullineux, J. Lond. Math. Soc., Volume 54 (1996) no. 1, pp. 25-38 | DOI | MR | Zbl

[39] Kleshchev, Alexander Linear and projective representations of symmetric groups, Cambridge Tracts in Mathematics, 163, Cambridge University Press, 2005 | DOI | Zbl

[40] Kleshchev, Alexander; Mathas, Andrew; Ram, Arun Universal graded Specht modules for cyclotomic Hecke algebras, Proc. Lond. Math. Soc., Volume 105 (2012) no. 6, pp. 1245-1289 | DOI | MR | Zbl

[41] Kleshchev, Alexander; Ram, Arun Homogeneous representations of Khovanov–Lauda algebras, J. Eur. Math. Soc., Volume 12 (2010) no. 5, pp. 1293-1306 | DOI | MR | Zbl

[42] König, Steffen; Xi, Changchang Affine cellular algebras, Adv. Math., Volume 229 (2012) no. 1, pp. 139-182 | DOI | MR | Zbl

[43] Lascoux, Alain; Leclerc, Bernard; Thibon, Jean-Yves Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Commun. Math. Phys., Volume 181 (1996) no. 1, pp. 205-263 | DOI | MR | Zbl

[44] Lauda, Aaron D.; Vazirani, Monica Crystals from categorified quantum groups, Adv. Math., Volume 228 (2011) no. 2, pp. 803-861 | DOI | MR | Zbl

[45] Li, Ge Integral basis theorem of cyclotomic Khovanov–Lauda–Rouquier algebras of type A, J. Algebra, Volume 482 (2017), pp. 1-101 | DOI | MR | Zbl

[46] Lusztig, George Canonical bases arising from quantized enveloping algebras, J. Am. Math. Soc., Volume 3 (1990) no. 2, pp. 447-498 | DOI | MR | Zbl

[47] Lusztig, George Introduction to quantum groups, Progress in Mathematics, 110, Birkhäuser, 1993 | Zbl

[48] Lyle, Sinéad; Mathas, Andrew Blocks of cyclotomic Hecke algebras, Adv. Math., Volume 216 (2007) no. 2, pp. 854-878 | DOI | MR | Zbl

[49] Mathas, Andrew Iwahori–Hecke algebras and Schur algebras of the symmetric group, University Lecture Series, 15, American Mathematical Society, 1999 | DOI | Zbl

[50] Mathas, Andrew Matrix units and generic degrees for the Ariki–Koike algebras, J. Algebra, Volume 281 (2004) no. 2, pp. 695-730 | DOI | MR | Zbl

[51] Mathas, Andrew Seminormal forms and Gram determinants for cellular algebras, J. Reine Angew. Math., Volume 619 (2008), pp. 141-173 (with an appendix by Marcos Soriano) | DOI | MR | Zbl

[52] Mathas, Andrew Cyclotomic quiver Hecke algebras of type A, Modular representation theory of finite and $p$ -adic groups (Teck, Gan Wee; Tan, Kai Meng, eds.) (National University of Singapore Lecture Notes Series), Volume 30, World Scientific, 2015, pp. 165-266 | DOI | MR | Zbl

[53] Mathas, Andrew Restricting Specht modules of cyclotomic Hecke algebras, Sci. China, Math., Volume 61 (2018), pp. 299-310 (Special Issue on Representation Theory) | DOI | MR | Zbl

[54] Mathas, Andrew Computing graded decomposition numbers of KLR algebras (2022), p. in preparation

[55] Mathas, Andrew Intertwiners and Garnir relations for KLR algebras (2023), p. in preparation

[56] Mathas, Andrew; Tubbenhauer, Daniel Cellularity and subdivision of KLR and weighted KLRW algebras, Math. Ann. (2023) (online first) | DOI

[57] Mathas, Andrew; Tubbenhauer, Daniel Cellularity for weighted KLRW algebras of types $B$ , $A^{(2)}$ , $D^{(2)}$ , J. Lond. Math. Soc., Volume 107 (2023) no. 3, pp. 1002-1044 | DOI | MR | Zbl

[58] Misra, Kailash; Miwa, Tetsuji Crystal base for the basic representation of $U_{q} (𝔰𝔩 (n))$ , Commun. Math. Phys., Volume 134 (1990) no. 1, pp. 79-88 | DOI | MR | Zbl

[59] Mullineux, Glen Bijections of $p$ -regular partitions and $p$ -modular irreducibles of the symmetric groups, J. Lond. Math. Soc., Volume 20 (1979) no. 1, pp. 60-66 | DOI | MR | Zbl

[60] Năstăsescu, Constantin; Van Oystaeyen, Freddy Methods of graded rings, Lecture Notes in Mathematics, 1836, Springer, 2004 | DOI | Zbl

[61] Premat, Alejandra Fock space representations and crystal bases for $C_{n}^{(1)}$ , J. Algebra, Volume 278 (2004) no. 1, pp. 227-241 | DOI | MR | Zbl

[62] Rouquier, Raphaël 2-Kac-Moody algebras (2008) | arXiv

[63] Rouquier, Raphaël Quiver Hecke algebras and 2-Lie algebras, Algebra Colloq., Volume 19 (2012) no. 2, pp. 359-410 | DOI | MR | Zbl

[64] Ryom-Hansen, Steen Grading the translation functors in type A, J. Algebra, Volume 274 (2004) no. 1, pp. 138-163 | DOI | MR | Zbl

[65] Speyer, Liron On the semisimplicity of the cyclotomic quiver Hecke algebra of type $C$ , Proc. Am. Math. Soc., Volume 146 (2018) no. 5, pp. 1845-1857 | DOI | MR | Zbl

[66] Stein, William A. et al. Sage Mathematics Software, 2016 (http://www.sagemath.org)

[67] Van Geel, Jan; Van Oystaeyen, Freddy About graded fields, Indag. Math., New Ser., Volume 43 (1981) no. 3, pp. 273-286 | DOI | MR | Zbl

[68] Varagnolo, Michela; Vasserot, Eric Canonical bases and KLR-algebras, J. Reine Angew. Math., Volume 659 (2011), pp. 67-100 | DOI | MR | Zbl

[69] Webster, Ben Knot invariants and higher representation theory, Memoirs of the American Mathematical Society, 1191, American Mathematical Society, 2017 | DOI | Zbl

[70] Webster, Ben Rouquier’s conjecture and diagrammatic algebra, Forum Math. Sigma, Volume 5 (2017), Paper no. 27 | DOI | MR | Zbl

[71] Webster, Ben Weighted Khovanov–Lauda–Rouquier algebras, Doc. Math., Volume 24 (2019), pp. 209-250 | DOI | MR | Zbl

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