## Algebra & Number Theory

### Quantized mixed tensor space and Schur–Weyl duality

#### Abstract

Let $R$ be a commutative ring with $1$ and $q$ an invertible element of $R$. The (specialized) quantum group $U=Uq(gln)$ over $R$ of the general linear group acts on mixed tensor space $V⊗r⊗V∗⊗s$, where $V$ denotes the natural $U$-module $Rn$, $r$ and $s$ are nonnegative integers and $V∗$ is the dual $U$-module to $V$. The image of $U$ in $EndR(V⊗r⊗V∗⊗s)$ is called the rational $q$-Schur algebra $Sq(n;r,s)$. We construct a bideterminant basis of $Sq(n;r,s)$. There is an action of a $q$-deformation $Br,sn(q)$ of the walled Brauer algebra on mixed tensor space centralizing the action of $U$. We show that $EndBr,sn(q)(V⊗r⊗V∗⊗s)=Sq(n;r,s)$. By a previous result, the image of $Br,sn(q)$ in $EndR(V⊗r⊗V∗⊗s)$ is $EndU(V⊗r⊗V∗⊗s)$. Thus, a mixed tensor space as $(U,Br,sn(q))$-bimodule satisfies Schur–Weyl duality.

#### Article information

Source
Algebra Number Theory, Volume 7, Number 5 (2013), 1121-1146.

Dates
Revised: 12 April 2012
Accepted: 20 June 2012
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513730007

Digital Object Identifier
doi:10.2140/ant.2013.7.1121

Mathematical Reviews number (MathSciNet)
MR3101074

Zentralblatt MATH identifier
1290.17012

#### Citation

Dipper, Richard; Doty, Stephen; Stoll, Friederike. Quantized mixed tensor space and Schur–Weyl duality. Algebra Number Theory 7 (2013), no. 5, 1121--1146. doi:10.2140/ant.2013.7.1121. https://projecteuclid.org/euclid.ant/1513730007

#### References

• G. Benkart, M. Chakrabarti, T. Halverson, R. Leduc, C. Lee, and J. Stroomer, “Tensor product representations of general linear groups and their connections with Brauer algebras”, J. Algebra 166:3 (1994), 529–567.
• J. Brundan and C. Stroppel, “Gradings on walled Brauer algebras and Khovanov's arc algebras”, preprint, 2011.
• R. Dipper and S. Donkin, “Quantum ${\rm GL}\sb n$”, Proc. London Math. Soc. $(3)$ 63:1 (1991), 165–211.
• R. Dipper and S. Doty, “The rational Schur algebra”, Represent. Theory 12 (2008), 58–82.
• R. Dipper and G. James, “The $q$-Schur algebra”, Proc. London Math. Soc. $(3)$ 59:1 (1989), 23–50.
• R. Dipper, S. Doty, and F. Stoll, “The quantized walled Brauer algebra and mixed tensor space”, preprint, 2012.
• K. R. Goodearl, “Commutation relations for arbitrary quantum minors”, Pacific J. Math. 228:1 (2006), 63–102.
• R. M. Green, “$q$-Schur algebras as quotients of quantized enveloping algebras”, J. Algebra 185:3 (1996), 660–687.
• J. Hong and S.-J. Kang, Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics 42, American Mathematical Society, Providence, RI, 2002.
• R. Q. Huang and J. J. Zhang, “Standard basis theorem for quantum linear groups”, Adv. Math. 102:2 (1993), 202–229.
• J. C. Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics 6, American Mathematical Society, Providence, RI, 1996.
• K. Koike, “On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters”, Adv. Math. 74:1 (1989), 57–86.
• M. Kosuda and J. Murakami, “Centralizer algebras of the mixed tensor representations of quantum group $U\sb q({\rm gl}(n,{\bf C}))$”, Osaka J. Math. 30:3 (1993), 475–507.
• R. E. Leduc, A two-parameter version of the centralizer algebra of the mixed tensor representations of the general linear group and quantum general linear group, Ph.D. thesis, University of Wisconsin–Madison, 1994.
• G. Lusztig, “Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra”, J. Amer. Math. Soc. 3:1 (1990), 257–296.
• I. Schur, “Über die rationalen Darstellungen der allgemeinen linearen Gruppe”, Sitzungsber. Akad. Berlin (1927), 58–75. Reprinted as pp. 68–85 in Gesammelte Abhandlungen, III, Springer, Berlin, 1973.
• J. R. Stembridge, “Rational tableaux and the tensor algebra of ${\rm gl}\sb n$”, J. Combin. Theory Ser. A 46:1 (1987), 79–120.
• R. Tange, “A bideterminant basis for a reductive monoid”, J. Pure Appl. Algebra 216:5 (2012), 1207–1221.
• V. G. Turaev, “Operator invariants of tangles, and $R$-matrices”, Izv. Akad. Nauk SSSR Ser. Mat. 53:5 (1989), 1073–1107, 1135. In Russian; translated in Math. USSR, Izv. 35:2 (1990), 411–444.