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.

Revised:

Accepted:

Published online:

Keywords: Cyclotomic KLR algebras, quiver Hecke algebras, categorification, quantum groups, representation theory, cellular algebras, Specht modules, seminormal forms

^{1}; Mathas, Andrew

^{2}

@article{ART_2024__1_2_193_0, author = {Evseev, Anton and Mathas, Andrew}, title = {Content systems and deformations of cyclotomic {KLR} algebras of type $A$ and $C$}, journal = {Annals of Representation Theory}, pages = {193--297}, publisher = {The Publishers of ART}, volume = {1}, number = {2}, year = {2024}, doi = {10.5802/art.8}, language = {en}, url = {https://art.centre-mersenne.org/articles/10.5802/art.8/} }

TY - JOUR AU - Evseev, Anton AU - Mathas, Andrew TI - Content systems and deformations of cyclotomic KLR algebras of type $A$ and $C$ JO - Annals of Representation Theory PY - 2024 SP - 193 EP - 297 VL - 1 IS - 2 PB - The Publishers of ART UR - https://art.centre-mersenne.org/articles/10.5802/art.8/ DO - 10.5802/art.8 LA - en ID - ART_2024__1_2_193_0 ER -

%0 Journal Article %A Evseev, Anton %A Mathas, Andrew %T Content systems and deformations of cyclotomic KLR algebras of type $A$ and $C$ %J Annals of Representation Theory %D 2024 %P 193-297 %V 1 %N 2 %I The Publishers of ART %U https://art.centre-mersenne.org/articles/10.5802/art.8/ %R 10.5802/art.8 %G en %F ART_2024__1_2_193_0

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/

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

[5] Representation type of finite quiver Hecke algebras of type ${A}_{2\ell}^{\left(2\right)}$, J. Algebra, Volume 397 (2014), pp. 457-488 | DOI | MR | Zbl

[6] Representation type of finite quiver Hecke algebras of type ${C}_{l}^{\left(1\right)}$, Osaka J. Math., Volume 53 (2016) no. 2, pp. 463-488 | Zbl

[7] Representation type of finite quiver Hecke algebras of type ${D}_{\ell +1}^{\left(2\right)}$, Trans. Am. Math. Soc., Volume 368 (2016) no. 5, pp. 3211-3242 | DOI | MR | Zbl

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

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

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

[13] Graded Specht modules, J. Reine Angew. Math., Volume 655 (2011), pp. 61-87 | DOI | MR | Zbl

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

[16] Derived equivalences for symmetric groups and ${\mathrm{\U0001d530\U0001d529}}_{2}$-categorification, Ann. Math., Volume 167 (2008) no. 1, pp. 245-298 | DOI | MR | Zbl

[17] Some graded decomposition matrices of finite quiver Hecke algebras in type C (2023), p. in preparation

[18] Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, 11, Interscience Publishers, 1962 | Zbl

[19] Cyclotomic $q$-Schur algebras, Math. Z., Volume 229 (1998) no. 3, pp. 385-416 | DOI | MR | Zbl

[20] 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] Cellular algebras, Invent. Math., Volume 123 (1996) no. 1, pp. 1-34 | DOI | MR | Zbl

[22] Affine ${\widehat{sl}}_{p}$ controls the modular representation theory of the symmetric group and related algebras, 1999, p. preprint

[23] $q$-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] 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] Graded induction for Specht modules, Int. Math. Res. Not., Volume 2012 (2012) no. 6, pp. 1230-1263 | DOI | MR | Zbl

[26] 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] 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] Graded dimensions and monomial bases for the cyclotomic quiver Hecke algebras (2024) (to appear in Communications in Contemporary Mathematics) | arXiv | DOI

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

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

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

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

[35] 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] A diagrammatic approach to categorification of quantum groups. I, Represent. Theory, Volume 13 (2009), pp. 309-347 | DOI | MR | Zbl

[37] Crystal bases and generalized Lascoux–Leclerc–Thibon (LLT) algorithm for the quantum affine algebra ${U}_{q}\left({C}_{n}^{\left(1\right)}\right)$, J. Math. Phys., Volume 45 (2004) no. 12, pp. 4878-4895 | DOI | MR | Zbl

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

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

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

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

[43] 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] Crystals from categorified quantum groups, Adv. Math., Volume 228 (2011) no. 2, pp. 803-861 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

[58] Crystal base for the basic representation of ${U}_{q}\left(\mathrm{\U0001d530\U0001d529}\left(n\right)\right)$, Commun. Math. Phys., Volume 134 (1990) no. 1, pp. 79-88 | DOI | MR | Zbl

[59] 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] Methods of graded rings, Lecture Notes in Mathematics, 1836, Springer, 2004 | DOI | Zbl

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

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

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

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

[65] 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] et al. Sage Mathematics Software, 2016 (http://www.sagemath.org)

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

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

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

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

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

*Cited by Sources: *