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Seminormal forms and cyclotomic quiver Hecke algebras of type A

Abstract

This paper shows that the cyclotomic quiver Hecke algebras of type A, and the gradings on these algebras, are intimately related to the classical seminormal forms. We start by classifying all seminormal bases and then give an explicit “integral” closed formula for the Gram determinants of the Specht modules in terms of the combinatorics associated with the KLR grading. We then use seminormal forms to give a deformation of the KLR algebras of type A. This makes it possible to study the cyclotomic quiver Hecke algebras in terms of the semisimple representation theory and seminormal forms. As an application we construct a new distinguished graded cellular basis of the cyclotomic KLR algebras of type A.

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References

  1. 1.

    Ariki, S.: On the semi-simplicity of the Hecke algebra of \((\mathbb{Z}/r\mathbb{Z})\wr \mathfrak{S}_n\). J. Algebra 169, 216–225 (1994)

  2. 2.

    Ariki, S.: On the classification of simple modules for cyclotomic Hecke algebras of type \(G(m,1, n)\) and Kleshchev multipartitions. Osaka J. Math. 38, 827–837 (2001)

  3. 3.

    Ariki, S., Koike, K.: A Hecke algebra of \(({ Z}/r{ Z})\wr {\mathfrak{S}}_n\) and construction of its irreducible representations. Adv. Math. 106, 216–243 (1994)

  4. 4.

    Ariki, S., Mathas, A., Rui, H.: Cyclotomic Nazarov-Wenzl algebras. Nagoya Math. J. 182, 47–134 (2006) (Special issue in honour of George Lusztig). arXiv:math/0506467

  5. 5.

    Brundan, J., Kleshchev, A.: Schur–Weyl duality for higher levels. Sel. Math. (N.S.) 14, 1–57 (2008)

  6. 6.

    Brundan, J., Kleshchev, A.: Blocks of cyclotomic Hecke algebras and Khovanov–Lauda algebras. Invent. Math. 178, 451–484 (2009)

  7. 7.

    Brundan, J., Kleshchev, A.: Graded decomposition numbers for cyclotomic Hecke algebras. Adv. Math. 222, 1883–1942 (2009)

  8. 8.

    Brundan, J., Kleshchev, A., Wang, W.: Graded Specht modules. J. Reine Angew. Math. 655, 61–87 (2011). arXiv:0901.0218

  9. 9.

    Brundan, J., Stroppel, C.: Highest weight categories arising from Khovanov’s diagram algebra III: category \({\cal O}\). Represent. Theory 15, 170–243 (2011). arXiv:0812.1090

  10. 10.

    Dipper, R., James, G., Mathas, A.: Cyclotomic \(q\)-Schur algebras. Math. Z. 229, 385–416 (1998)

  11. 11.

    Dipper, R., Mathas, A.: Morita equivalences of Ariki–Koike algebras. Math. Z. 240, 579–610 (2002)

  12. 12.

    Graham, J.J., Lehrer, G.I.: Cellular algebras. Invent. Math. 123, 1–34 (1996)

  13. 13.

    Hoffnung, A.E., Lauda, A.D.: Nilpotency in type \(A\) cyclotomic quotients. J. Algebraic Comb. 32, 533–555 (2010)

  14. 14.

    Hu, J., Mathas, A.: Graded cellular bases for the cyclotomic Khovanov–Lauda–Rouquier algebras of type \(A\). Adv. Math. 225, 598–642 (2010). arXiv:0907.2985

  15. 15.

    Hu, J., Mathas, A.: Cyclotomic quiver Schur algebras for linear quivers. Proc. Lond. Math. Soc. 110, 1315–1386 (2015). arXiv:1110.1699

  16. 16.

    James, G., Mathas, A.: A \(q\)-analogue of the Jantzen–Schaper theorem. Proc. Lond. Math. Soc. (3) 74, 241–274 (1997)

  17. 17.

    Hu, J., Mathas, A.: The Jantzen sum formula for cyclotomic \(q\)-Schur algebras. Trans. Am. Math. Soc. 352, 5381–5404 (2000)

  18. 18.

    James, G., Murphy, G.E.: The determinant of the Gram matrix for a Specht module. J. Algebra 59, 222–235 (1979)

  19. 19.

    Kac, V.G.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)

  20. 20.

    Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53, 165–184 (1979)

  21. 21.

    Khovanov, M., Lauda, A.D.: A diagrammatic approach to categorification of quantum groups. I. Represent. Theory 13, 309–347 (2009)

  22. 22.

    Khovanov, M., Lauda, A.D.: A diagrammatic approach to categorification of quantum groups II. Trans. Am. Math. Soc. 363, 2685–2700 (2011)

  23. 23.

    Kleshchev, A., Mathas, A., Ram, A.: Universal graded Specht modules for cyclotomic Hecke algebras. Proc. Lond. Math. Soc. (3) 105, 1245–1289 (2012). arXiv:1102.3519

  24. 24.

    Li, G.: Integral Basis Theorem of Cyclotomic Khovanov–Lauda–Rouquier Algebras of Type A. Ph.D. thesis, University of Sydney (2012)

  25. 25.

    Maksimau, R.: Quiver Schur algebras and Koszul duality. J. Algebra 406, 91–133 (2014). arXiv:1307.6013

  26. 26.

    Mathas, A.: Iwahori–Hecke algebras and Schur algebras of the symmetric group. In: University Lecture Series, vol. 15. American Mathematical Society, Providence (1999)

  27. 27.

    Mathas, A.: Matrix units and generic degrees for the Ariki–Koike algebras. J. Algebra 281, 695–730 (2004). arXiv:math/0108164

  28. 28.

    Mathas, A.: Seminormal forms and Gram determinants for cellular algebras. J. Reine Angew. Math. 619, 141–173 (2008) (With an appendix by Marcos Soriano). arXiv:math/0604108

  29. 29.

    Murphy, G.E.: The idempotents of the symmetric group and Nakayama’s conjecture. J. Algebra 81, 258–265 (1983)

  30. 30.

    Okounkov, A., Vershik, A.: A new approach to representation theory of symmetric groups. Sel. Math. (N.S.) 2, 581–605 (1996)

  31. 31.

    Rouquier, R.: 2-Kac–Moody algebras (2008, preprint). arXiv:0812.5023

  32. 32.

    Rouquier, R., Shan, P., Varagnolo, M., Vasserot, E.: Categorifications and cyclotomic rational double affine Hecke algebras (2013, preprint). arXiv:1305.4456

  33. 33.

    Ryom-Hansen, S.: The Schaper formula and the Lascoux, Leclerc and Thibon algorithm. Lett. Math. Phys. 64, 213–219 (2003)

  34. 34.

    Ryom-Hansen, S.: Young’s seminormal form and simple modules for \(S_n\) in characteristic \(p\), 2011. Algebras Represent. Theory 16, 15871609 (2013). arXiv:1107.3076

  35. 35.

    Serre, J.-P.: Local fields. In: Graduate Texts in Mathematics, vol. 67. Springer, New York (1979) (Translated from the French by Marvin Jay Greenberg)

  36. 36.

    Stroppel, C., Webster, B.: Quiver Schur algebras and \(q\)-Fock space (2011, preprint). arXiv:1110.1115

  37. 37.

    Young, A.: On quantitative substitutional analysis I. Proc. Lond. Math. Soc. 33, 97–145 (1900)

  38. 38.

    Yvonne, X.: A conjecture for \(q\)-decomposition matrices of cyclotomic \(v\)-Schur algebras. J. Algebra 304, 419–456 (2006)

Download references

Acknowledgments

J. Hu and A. Mathas were supported by the Australian Research Council. J. Hu author was also supported by the National Natural Science Foundation of China.

Author information

Correspondence to Andrew Mathas.

Appendix: Seminormal forms for the linear quiver

Appendix: Seminormal forms for the linear quiver

In this appendix we show how the results in this paper work when \(e=0\) so that \(\xi \in K\) is either not a root of unity or \(\xi =1\) and K is a field of characteristic zero. In order to define a modular system we have to leave the case where the cyclotomic parameters \(Q_1,\ldots ,Q_\ell \) are integral, that is, when \(Q_l=[\kappa _l]\) for \(1\le l\le \ell \). This causes quite a few notational inconveniences, but otherwise the story is much the same as for the case when \(e>0\). We do not develop the full theory of “0-idempotent subrings” here. Rather, we show just one way of proving the results in this paper when \(e=0\).

Fix a field K and \(0\ne \xi \in K\) of quantum characteristic e. That is, either \(\xi =1\) and K is a field of characteristic zero or \(\xi ^d\ne 1\) for \(d\in \mathbb {Z}\). The multicharge \({\varvec{\kappa }}\in \mathbb {Z}^\ell \) is arbitrary.

Let \({\mathcal {O}}=\mathbb {Z}[x,\xi ]_{(x)}\) be the localisation of \(\mathbb {Z}[x,\xi ]\) at the principal ideal generated by x. Let \(\fancyscript{K}=\mathbb {Q}(x,\xi )\) be the field of fractions of \({\mathcal {O}}\). Define \(\mathcal {H}^\varLambda _n({\mathcal {O}})\) to be the cyclotomic Hecke algebra of type A with Hecke parameter \(t=\xi \), a unit in \({\mathcal {O}}\), and cyclotomic parameters

$$\begin{aligned} Q_l=x^l+[\kappa _l],\quad \text {for }1\le l\le \ell , \end{aligned}$$

where, as before, \([k]=[k]_t\) for \(k\in \mathbb {Z}\). Then \(\mathcal {H}^\varLambda _{n}(\fancyscript{K})=\mathcal {H}^\varLambda _n({\mathcal {O}})\otimes _{\mathcal {O}}\fancyscript{K}\) is split semisimple in view of Ariki’s semisimplicity condition [1]. Moreover, by definition, \(\mathcal {H}^\varLambda _{n}(K)\cong \mathcal {H}^\varLambda _n({\mathcal {O}})\otimes _{\mathcal {O}}K\), where we consider K as an \({\mathcal {O}}\)-module by setting x act on K as multiplication by zero.

Define a new content function for \(\mathcal {H}^\varLambda _n({\mathcal {O}})\) by setting

$$\begin{aligned} C_\gamma =t^{c-r}x^{l}+[\kappa _l+c-r], \end{aligned}$$

for a node \(\gamma =(l,r,c)\). We will also need the previous definition of contents below. If \({\mathfrak t}\in {\mathop {\mathrm{Std}}\nolimits }(\mathcal {P}^{\varLambda }_{n})\) is a tableau and \(1\le k\le n\) then set \(C_k({\mathfrak t})=C_\gamma \), where \(\gamma \) is the unique node such that \({\mathfrak t}(\gamma )=k\).

As in Sect. 2.5, let \(\{m_{{\mathfrak s}{\mathfrak t}}\,|\,({\mathfrak s},{\mathfrak t})\in {\mathrm{Std}}^2(\mathcal {P}^{\varLambda }_{n})\}\) be the Murphy basis of \(\mathcal {H}^\varLambda _n({\mathcal {O}})\). Then the analogue of Lemma 2.6 is that if \(1\le r\le n\) then

$$\begin{aligned} m_{{\mathfrak s}{\mathfrak t}}L_r=C_r({\mathfrak t})m_{{\mathfrak s}{\mathfrak t}}+\sum _{({\mathfrak u},{\mathfrak v})\vartriangleright ({\mathfrak s},{\mathfrak t})}r_{{\mathfrak u}{\mathfrak v}}m_{{\mathfrak u}{\mathfrak v}}, \end{aligned}$$

for some \(r_{{\mathfrak u}{\mathfrak v}}\in {\mathcal {O}}\). As in Sect. 3.1 define a \(*\)-seminormal basis of \(\mathcal {H}^\varLambda _{n}(\fancyscript{K})\) to be a basis \(\{f_{{\mathfrak s}{\mathfrak t}}\}\) of simultaneous two-sided eigenvectors for \(L_1,\ldots ,L_n\) such that \(f_{{\mathfrak s}{\mathfrak t}}^{*}=f_{{\mathfrak t}{\mathfrak s}}\).

Define a seminormal coefficient system for \(\mathcal {H}^\varLambda _n({\mathcal {O}})\) to be a set of scalars \({\varvec{\alpha }}=\{\alpha _r({\mathfrak s})\}\) satisfying Definition 3.5(a), (b) and such that if \({\mathfrak s}\in {\mathop {\mathrm{Std}}\nolimits }(\mathcal {P}^{\varLambda }_{n})\) and \({\mathfrak u}={\mathfrak s}(r,r+1)\in {\mathop {\mathrm{Std}}\nolimits }(\mathcal {P}^{\varLambda }_{n})\) then

$$\begin{aligned} \alpha _r({\mathfrak s})\alpha _r({\mathfrak u}) = \frac{(1-C_r({\mathfrak s})+tC_r({\mathfrak u}))(1+tC_r({\mathfrak s})-C_r({\mathfrak u}))}{P_r({\mathfrak s})P_r({\mathfrak u})}, \end{aligned}$$
(A1)

where \(P_r({\mathfrak s})=C_r({\mathfrak u})-C_r({\mathfrak s})\), and where \(\alpha _r({\mathfrak s})=0\) if \({\mathfrak u}\notin {\mathop {\mathrm{Std}}\nolimits }(\mathcal {P}^{\varLambda }_{n})\).

As in Theorem 3.9, each seminormal basis of \(\mathcal {H}^\varLambda _n(\fancyscript{K})\) is determined by a seminormal coefficient system \({\varvec{\alpha }}=\{\alpha _r({\mathfrak s})\}\), such that

$$\begin{aligned} T_rf_{{\mathfrak s}{\mathfrak t}}=\alpha _r({\mathfrak s})f_{{\mathfrak u}{\mathfrak t}}+\frac{1+(t-1)C_{r+1}({\mathfrak s})}{P_r({\mathfrak s})}f_{{\mathfrak s}{\mathfrak t}}, \quad \text {where }{\mathfrak u}={\mathfrak s}(r,r+1), \end{aligned}$$

together with a set of scalars \(\{\gamma _{{\mathfrak t}^{\varvec{\lambda }}}\,|\,{\varvec{\lambda }}\in \mathcal {P}^{\varLambda }_{n}\}\). Notice that \(I=\mathbb {Z}\), since \(e=0\), so if \(\mathbf {i}\in I^n\) then \({\mathfrak t}\in {\mathop {\mathrm{Std}}\nolimits }(\mathbf {i})\) if and only if \(c_r({\mathfrak t})=i_r\) and, in turn, this is equivalent to the constant term of \(C_r({\mathfrak t})\) being equal to \([i_r]\), for \(1\le r\le n\). Arguing as in Lemma 4.3,

$$\begin{aligned} f_{\mathbf {i}}^{\mathcal {O}}= \sum _{{\mathfrak t}\in {\mathop {\mathrm{Std}}\nolimits }(\mathbf {i})}\frac{1}{\gamma _{\mathfrak t}}f_{{\mathfrak t}{\mathfrak t}}\in \mathcal {H}^\varLambda _n({\mathcal {O}}). \end{aligned}$$

With these definitions in place all of the arguments in Sect. 4 go through with only minor changes. In particular, if \(1\le r\le n\) and \(\mathbf {i}\in I^n\) then Definition 4.12 should be replaced by

$$\begin{aligned} \psi ^{\mathcal {O}}_rf_{\mathbf {i}}^{\mathcal {O}}={\left\{ \begin{array}{ll} (T_r+1)\frac{1}{M_r}f_{\mathbf {i}}^{\mathcal {O}},&{}\text {if }i_r=i_{r+1}\\ (T_rL_r-L_rT_r)f_{\mathbf {i}}^{\mathcal {O}},&{}\text {if }i_r=i_{r+1}+1,\\ (T_rL_r-L_rT_r)\frac{1}{M_r}f_{\mathbf {i}}^{\mathcal {O}},&{}\text {otherwise,}\\ \end{array}\right. } \end{aligned}$$

and \(y^{{\mathcal {O}}}_rf_{\mathbf {i}}^{\mathcal {O}}=\big (L_r-C_r({\mathfrak t})\big )f_{\mathbf {i}}^{\mathcal {O}}\) where, as before, \(M_r=1-L_r+tL_{r+1}\). With these new definitions, if \({\mathfrak s}\in {\mathop {\mathrm{Std}}\nolimits }(\mathbf {i})\), for \(\mathbf {i}\in I^m\), and \(1\le r\le n\) then Lemma 4.19 becomes

$$\begin{aligned} \psi ^{\mathcal {O}}_r f_{{\mathfrak s}{\mathfrak t}}=B_r({\mathfrak s})f_{{\mathfrak s}{\mathfrak t}}+\frac{\delta _{i_ri_{r+1}}}{P_r({\mathfrak s})}f_{{\mathfrak u}{\mathfrak t}}, \end{aligned}$$

where \({\mathfrak u}={\mathfrak s}(r,r+1)\) and

$$\begin{aligned} B_r({\mathfrak s})={\left\{ \begin{array}{ll} \frac{\alpha _r({\mathfrak s})}{1-C_r({\mathfrak s})+tC_{r+1}({\mathfrak s})},&{}\text {if }i_r=i_{r+1},\\ \alpha _r({\mathfrak s})P_r({\mathfrak s}),&{}\text {if }i_r=i_{r+1}+1,\\ \frac{\alpha _r(s)P_r({\mathfrak s})}{1-C_r({\mathfrak s})+tC_{r+1}({\mathfrak s})},&{}\text {otherwise.} \end{array}\right. } \end{aligned}$$

Observe that if \({\mathfrak u}={\mathfrak s}(r,r+1)\) is a standard tableau then, using (A1), the definitions imply that

$$\begin{aligned} B_r({\mathfrak s})B_r({\mathfrak u}) = {\left\{ \begin{array}{ll} \frac{1}{P_r({\mathfrak s})P_r({\mathfrak u})},&{}\text {if }i_r=i_{r+1},\\ (1-C_r({\mathfrak s})+tC_r({\mathfrak u}))(1+tC_r({\mathfrak s})-C_r({\mathfrak u})),&{}\text {if }i_r\leftrightarrows i_{r+1},\\ (1+tC_r({\mathfrak s})-C_r({\mathfrak u})),&{}\text {if }i_r\rightarrow i_{r+1},\\ (1-C_r({\mathfrak s})+tC_r({\mathfrak u})),&{}\text {if }i_r\leftarrow i_{r+1},\\ 1,&{}\text {otherwise.} \end{array}\right. } \end{aligned}$$

Comparing this with Lemma 4.22, it is now easy to see that analogues of Proposition 4.23 and Proposition 4.24 both hold in this situation. Hence, repeating the arguments of Sect. 4.4, a suitable modification of Theorem A also holds. Similarly, the construction of the bases in Sects. 5 and Sect. 6 now goes though largely without change.

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Hu, J., Mathas, A. Seminormal forms and cyclotomic quiver Hecke algebras of type A . Math. Ann. 364, 1189–1254 (2016). https://doi.org/10.1007/s00208-015-1242-8

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Mathematics Subject Classification

  • 20G43
  • 20C08
  • 20C30