Presenting affine Schur algebras

Fu, Qiang; Liu, Mingqiang

doi:10.1090/tran/7451

Presenting affine Schur algebras
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by Qiang Fu and Mingqiang Liu PDF

Trans. Amer. Math. Soc. 371 (2019), 5487-5503 Request permission

Abstract:

The universal enveloping algebra $\mathcal {U}(\widehat {\frak {gl}}_n)$ of $\widehat {\frak {gl}}_n$ was realized in [A double Hall algebra approach to affine quantum Schur–Weyl theory, Cambridge University Press, Cambridge, 2012, Ch. 6] using affine Schur algebras. In particular some explicit multiplication formulas in affine Schur algebras were derived. We use these formulas to study the structure of affine Schur algebras. In particular, we give a presentation of the affine Schur algebra ${\mathcal S}_{{\!\vartriangle }}(n,r)_{\mathbb Q}$ over $\mathbb {Q}$.

References

A. A. Beilinson, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformation of $\textrm {GL}_n$, Duke Math. J. 61 (1990), no. 2, 655–677. MR 1074310, DOI 10.1215/S0012-7094-90-06124-1
Bangming Deng, Jie Du, and Qiang Fu, A double Hall algebra approach to affine quantum Schur-Weyl theory, London Mathematical Society Lecture Note Series, vol. 401, Cambridge University Press, Cambridge, 2012. MR 3113018, DOI 10.1017/CBO9781139226660
Stephen Doty and Anthony Giaquinto, Presenting Schur algebras, Int. Math. Res. Not. 36 (2002), 1907–1944. MR 1920169, DOI 10.1155/S1073792802201026
S. R. Doty and R. M. Green, Presenting affine $q$-Schur algebras, Math. Z. 256 (2007), no. 2, 311–345. MR 2289877, DOI 10.1007/s00209-006-0076-1
Jie Du, A note on quantized Weyl reciprocity at roots of unity, Algebra Colloq. 2 (1995), no. 4, 363–372. MR 1358684
Jie Du and Qiang Fu, A modified BLM approach to quantum affine $\mathfrak {gl}_n$, Math. Z. 266 (2010), no. 4, 747–781. MR 2729289, DOI 10.1007/s00209-009-0596-6
Jie Du and Qiang Fu, Quantum affine $\mathfrak {gl}_n$ via Hecke algebras, Adv. Math. 282 (2015), 23–46. MR 3374521, DOI 10.1016/j.aim.2015.06.007
J. Du and Q. Fu, The integral quantum loop algebra of $\mathfrak {gl}_n$, Internat. Math. Res. Notices, https://doi.org/10.1093/imrn/rnx300, in press.
Jie Du and Brian Parshall, Monomial bases for $q$-Schur algebras, Trans. Amer. Math. Soc. 355 (2003), no. 4, 1593–1620. MR 1946407, DOI 10.1090/S0002-9947-02-03188-4
Qiang Fu, On Schur algebras and little Schur algebras, J. Algebra 322 (2009), no. 5, 1637–1652. MR 2543627, DOI 10.1016/j.jalgebra.2009.04.027
Qiang Fu, Integral affine Schur-Weyl reciprocity, Adv. Math. 243 (2013), 1–21. MR 3062738, DOI 10.1016/j.aim.2013.03.025
Qiang Fu, BLM realization for the integral form of quantum $\mathfrak {gl}_n$, Commun. Contemp. Math. 17 (2015), no. 5, 1550019, 17. MR 3404750, DOI 10.1142/S0219199715500194
Qiang Fu, BLM realization for Frobenius-Lusztig kernels of type $A$, Math. Res. Lett. 23 (2016), no. 5, 1329–1350. MR 3601068, DOI 10.4310/MRL.2016.v23.n5.a4
Qiang Fu, BLM realization for $\mathcal U_{\Bbb Z}(\widehat {\mathfrak {gl}}_n)$, Commun. Contemp. Math. 20 (2018), no. 2, 1750013, 35. MR 3730749, DOI 10.1142/S0219199717500134
Victor Ginzburg and Éric Vasserot, Langlands reciprocity for affine quantum groups of type $A_n$, Internat. Math. Res. Notices 3 (1993), 67–85. MR 1208827, DOI 10.1155/S1073792893000078
R. M. Green, The affine $q$-Schur algebra, J. Algebra 215 (1999), no. 2, 379–411. MR 1686197, DOI 10.1006/jabr.1998.7753
G. Lusztig, Aperiodicity in quantum affine $\mathfrak {g}\mathfrak {l}_n$, Asian J. Math. 3 (1999), no. 1, 147–177. Sir Michael Atiyah: a great mathematician of the twentieth century. MR 1701926, DOI 10.4310/AJM.1999.v3.n1.a7
Marco Mackaay and Anne-Laure Thiel, A diagrammatic categorification of the affine $q$-Schur algebra $\widehat S(n,n)$ for $n\ge 3$, Pacific J. Math. 278 (2015), no. 1, 201–233. MR 3404672, DOI 10.2140/pjm.2015.278.201
Marco Mackaay and Anne-Laure Thiel, Categorifications of the extended affine Hecke algebra and the affine $q$-Schur algebra $\widehat {\mathbf S}(n,r)$ for $3\leq r<n$, Quantum Topol. 8 (2017), no. 1, 113–203. MR 3630283, DOI 10.4171/QT/88
Kevin McGerty, Generalized $q$-Schur algebras and quantum Frobenius, Adv. Math. 214 (2007), no. 1, 116–131. MR 2348025, DOI 10.1016/j.aim.2006.10.012