# Cellular structures using $\textbf{U}_q$-tilting modules

@article{Andersen2015CellularSU, title={Cellular structures using \$\textbf\{U\}\_q\$-tilting modules}, author={Henning Haahr Andersen and Catharina Stroppel and Daniel Tubbenhauer}, journal={arXiv: Quantum Algebra}, year={2015} }

We use the theory of $\textbf{U}_q$-tilting modules to construct cellular bases for centralizer algebras. Our methods are quite general and work for any quantum group $\textbf{U}_q$ attached to a Cartan matrix and include the non-semisimple cases for $q$ being a root of unity and ground fields of positive characteristic. Our approach also generalizes to certain categories containing infinite-dimensional modules. As applications, we give a new semisimplicty criterion for centralizer algebras…

## 19 Citations

Light ladders and clasp conjectures

- Mathematics
- 2015

Morphisms between tensor products of fundamental representations of the quantum group of sl(n) are described by the sl(n)-webs of Cautis-Kamnitzer-Morrison. Using these webs, we provide an explicit,…

Quivers for 𝑆𝐿₂ tilting modules

- Computer ScienceRepresentation Theory of the American Mathematical Society
- 2021

A quiver with relations depending on a prime is defined and the associated path algebra describes the category of tilting modules for <inline-formula content-type="math/mathml">.

THE CENTER OF SL2 TILTING MODULES

- MathematicsGlasgow Mathematical Journal
- 2021

Abstract In this note, we compute the centers of the categories of tilting modules for G = SL2 in prime characteristic, of tilting modules for the corresponding quantum group at a complex root of…

DIAGRAMMATIC CONSTRUCTION OF REPRESENTATIONS OF SMALL QUANTUM $$ \mathfrak{sl} $$2

- MathematicsTransformation Groups
- 2021

We provide a combinatorial description of the monoidal category generated by the fundamental representation of the small quantum group of $$ \mathfrak{sl} $$
sl
2 at a root of unity q of odd order.…

Schur–Weyl duality, Verma modules, and row
quotients of Ariki–Koike algebras

- Mathematics
- 2020

We prove a Schur-Weyl duality between the quantum enveloping algebra of $\mathfrak{gl}_m$ and certain quotient algebras of Ariki-Koike algebras, which we give explicitly. The duality involves several…

## References

SHOWING 1-10 OF 93 REFERENCES

Lectures on quantum groups

- Mathematics
- 1995

Introduction Gaussian binomial coefficients The quantized enveloping algebra $U_q(\mathfrak s \mathfrak {1}_2)$ Representations of $U_q(\mathfrak{sl}_2)$ Tensor products or: $U_q(\mathfrak{sl}_2)$ as…

Representations of algebraic groups

- Mathematics
- 1987

Part I. General theory: Schemes Group schemes and representations Induction and injective modules Cohomology Quotients and associated sheaves Factor groups Algebras of distributions Representations…

Graded cellular bases for Temperley–Lieb algebras of type A and B

- Mathematics
- 2012

We show that the Temperley–Lieb algebra of type A and the blob algebra (also known as the Temperley–Lieb algebra of type B) at roots of unity are $\mathbb{Z}$-graded algebras. We moreover show that…

SEMISIMPLICITY OF HECKE AND (WALLED) BRAUER ALGEBRAS

- MathematicsJournal of the Australian Mathematical Society
- 2016

We show how to use Jantzen’s sum formula for Weyl modules to prove semisimplicity criteria for endomorphism algebras of $\mathbf{U}_{q}$ -tilting modules (for any field $\mathbb{K}$ and any parameter…

Character formulas for tilting modules over Kac-Moody algebras

- Mathematics
- 1998

We show how to express the characters of tilting modules in a (possibly parabolic) category O over a Kac-Moody algebra in terms of the characters of simple highest weight modules. This settles, in…

Quiver Schur algebras for linear quivers

- Mathematics
- 2015

We define a graded quasi‐hereditary covering of the cyclotomic quiver Hecke algebras RnΛ of type A when e=0 (the linear quiver) or e>n . We prove that these algebras are quasi‐hereditary graded…

Highest Weight Categories Arising from Khovanov's Diagram Algebra I: Cellularity

- Mathematics
- 2011

This is the first of four articles studying some slight generalisations Hn m of Khovanov’s diagram algebra, as well as quasi-hereditary covers Kn m of these algebras in the sense of Rouquier, and…

Highest weight categories arising from Khovanov's diagram algebra III: category O

- Mathematics
- 2008

We prove that integral blocks of parabolic category O associated to the subalgebra gl(m) x gl(n) of gl(m+n) are Morita equivalent to quasi-hereditary covers of generalised Khovanov algebras. Although…