TY - JOUR

T1 - Degenerate 0-Schur algebras and Nil-Temperley-Lieb algebras

AU - Jensen, Bernt Tore

AU - Su, Xiuping

AU - Yang, Guiyu

N1 - Funding Information:
We would like to thank the referee for helpful comments, which have improved the exposition in this paper, and especially for pointing out the reference [15 ] and it?s relevance to our results in Section 6.
Publisher Copyright:
© 2019, The Author(s).

PY - 2020/6/30

Y1 - 2020/6/30

N2 - In Jensen and Su (J. Pure Appl. Algebra 219(2), 277–307 2014) constructed 0-Schur algebras, using double flag varieties. The construction leads to a presentation of 0-Schur algebras using quivers with relations and the quiver presentation naturally gives rise to a new class of algebras, which are introduced and studied in this paper. That is, these algebras are defined on the quivers of 0-Schur algebras with relations modified from the defining relations of 0-Schur algebras by a tuple of parameters t. In particular, when all the entries of t are 1, we recover 0-Schur algebras. When all the entries of t are zero, we obtain a class of basic algebras, which we call the degenerate 0-Schur algebras. We prove that the degenerate algebras are both associated graded algebras and quotients of 0-Schur algebras. Moreover, we give a geometric interpretation of the degenerate algebras using double flag varieties, in the same spirit as Jensen and Su (J. Pure Appl. Algebra 219(2), 277–307 2014), and show how the centralizer algebras are related to nil-Hecke and nil-Temperley-Lieb algebras.

AB - In Jensen and Su (J. Pure Appl. Algebra 219(2), 277–307 2014) constructed 0-Schur algebras, using double flag varieties. The construction leads to a presentation of 0-Schur algebras using quivers with relations and the quiver presentation naturally gives rise to a new class of algebras, which are introduced and studied in this paper. That is, these algebras are defined on the quivers of 0-Schur algebras with relations modified from the defining relations of 0-Schur algebras by a tuple of parameters t. In particular, when all the entries of t are 1, we recover 0-Schur algebras. When all the entries of t are zero, we obtain a class of basic algebras, which we call the degenerate 0-Schur algebras. We prove that the degenerate algebras are both associated graded algebras and quotients of 0-Schur algebras. Moreover, we give a geometric interpretation of the degenerate algebras using double flag varieties, in the same spirit as Jensen and Su (J. Pure Appl. Algebra 219(2), 277–307 2014), and show how the centralizer algebras are related to nil-Hecke and nil-Temperley-Lieb algebras.

KW - 0-Schur algebras

KW - Double flag varieties

KW - Nil-Hecke algebras

KW - Nil-Temperley-Lieb algebras

KW - Quivers

UR - http://www.scopus.com/inward/record.url?scp=85064242707&partnerID=8YFLogxK

U2 - 10.1007/s10468-019-09881-9

DO - 10.1007/s10468-019-09881-9

M3 - Article

SN - 1572-9079

VL - 23

SP - 1177

EP - 1196

JO - Algebra and Representation Theory

JF - Algebra and Representation Theory

IS - 3

ER -