Monomial base for little q-Schur algebra u _k(2, r) at even roots of unity

Abstract

Let u _k(2, r) be a little q-Schur algebra over k, where k is a field containing an l′-th primitive root ɛ of 1 with l′ ≥ 4 even, the author constructs a certain monomial base for little q-Schur algebra u _k(2, r).

Keywords

Little q-Schur algebra / Infinitesimal q-Schur algebra / Quantum group

References

[1] Beilinson A, Lusztig G, MacPherson R. A geometric setting for the quantum deformation of GLn. Duke Math. J., 1990, 61: 655-677

[2] Cox A. On some applications of infinitesimal methods to quantum groups and related algebras, 1997

[3] Cox A. On the blocks of the infinitesimal Schur algebras. Quart. J. Math., 2000, 51: 39-56

[4] Doty S, Erdmann K, Martin S, Nakano D. Representation type of Schur algebras. Math. Z., 1999, 232: 137-182

[5] Doty S, Giaquinto A. Presenting Schur algebras. International Mathematics Research Notices, 2002, 36: 1907-1944

[6] Doty S, Nakano D. Semisimple Schur algebras. Math. Proc. Cambridge Philos. Soc., 1998, 124: 15-20

[7] Doty S, Nakano D, Peters K. On Infinitesimal Schur algebras. Proc. London Math. Soc., 1996, 72: 588-612

[8] Doty S, Nakano D, Peters K. Infinitesimal Schur algebras of finite representation type. Quart. J. Math., 1997, 48: 323-345

[9] Du J. A note on the quantized Weyl reciprocity at roots of unity. Alg. Colloq., 1995, 2: 363-372

[10] Du J, Parshall B. Monomial bases for q-Schur algebras. Trans. Amer. Math. Soc., 2003, 355(4): 1593-1620

[11] Du J, Fu Q, Wang J P. Infinitesimal quantum gl_n and little q-Schur algebras. J. Algebra, 2005, 287: 199-233

[12] Du J, Fu Q, Wang J P. Representations of little q-Schur algebras. Pacific J. Math., 2012, 257(2): 343-378

[13] Erdmann K, Fu Q. Schur-Weyl duality for infinitesimal q-Schur algebras sq (2, r)₁. J. Algebra, 2008, 320: 1099-1114

[14] Fu Q. A comparison of infinitesimal and little q-Schur algebras. Comm. Algebra, 2005, 33: 2663-2682

[15] Fu Q. Little q-Schur algebras at even roots of unity. J. Algebra, 2007, 311: 202-215

[16] Fu Q. Finite representation type of infinitesimal q-Schur algebras. Pacific J. Math., 2008, 237(1): 57-76

[17] Fu Q. Semisimple infinitesimal q-Schur algebras. Arch. Math., 2008, 90: 295-303

[18] Fu Q. Tame representation type of infinitesimal q-Schur algebras. J. Algebra, 2008, 320: 369-386

[19] Fu Q. On Schur algebras and little Schur algebrass. J. Algebra, 2009, 322: 1637-1652

[20] Fu Q, Yang Q G. On the structure of End_{u_k (2)} (\Omega _k^{ \otimes r} ). J. Math. Phys., 2011, 52: 083507

[21] Lusztig G. Finite dimensional Hopf algebras arising from quantized universal enveloping algebras. J. Amer. Math. Soc., 1990, 3: 257-296

[22] Lusztig G. Canonical bases arising from quantized enveloping algebras. J. Amer. Math. Soc., 1990, 3: 447-498