Hecke algebras of finite type are cellular

Geck, Meinolf

doi:10.1007/s00222-007-0053-2

Cited by 76 publications

(109 citation statements)

References 24 publications

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“…Moreover if ℓ = 2 then {KZ h (∆ h (λ)) : λ ∈ P(2, n)} would be the set of cell modules with respect to the cellular algebra structure found on H q (W ) in [16 …”

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confidence: 99%

Quiver Varieties, Category for Rational Cherednik Algebras, and Hecke Algebras

Gordon¹

2010

International Mathematics Research Papers

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Abstract. We relate the representations of the rational Cherednik algebras associated with the complex reflection group µ ℓ ≀ Sn to sheaves on Nakajima quiver varieties associated with extended Dynkin graphs via a Z-algebra construction. This is done so that as the parameters defining the Cherednik algebra vary, the stability conditions defining the quiver variety change.This construction motivates us to use the geometry of the quiver varieties to interpret the ordering function (the c-function) used to define a highest weight structure on category O of the Cherednik algebra.This interpretation provides a natural partial ordering on O which we expect will respect the highest weight structure. This partial ordering has appeared in a conjecture of Yvonne on the composition factors in O and so our results provide a small step towards a geometric picture for that.We also interpret geometrically another ordering function (the a-function) used in the study of Hecke algebras. (The connection between Cherednik algebras and Hecke algebras is provided by the KZ-functor.) This is related to a conjecture of Bonnafé and Geck on equivalence classes of weight functions for Hecke algebras with unequal parameters since the classes should (and do for type B) correspond to the G.I.T. chambers defining the quiver varieties. As a result anything that can be defined via the quiver varieties, including the a-function, will be constant on these classes.

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“…Moreover if ℓ = 2 then {KZ h (∆ h (λ)) : λ ∈ P(2, n)} would be the set of cell modules with respect to the cellular algebra structure found on H q (W ) in [16 …”

mentioning

confidence: 99%

Quiver Varieties, Category for Rational Cherednik Algebras, and Hecke Algebras

Gordon¹

2010

International Mathematics Research Papers

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“…The bottom row of Figure 4 shows the average, minimum and maximum runtimes of GENERATE, REDUCE and CHECK tasks; the time recorded is GAP compute time, excluding communication and marshaling overheads. 8 We observe that CHECK tasks are generally expensive but regular, and REDUCE tasks are largely regular, with only some reps showing moderate irregularity (E 7 rep 48 is an outlier). However, GENERATE tasks are wildly irregular, varying by at least two orders of magnitude.…”

Section: Evaluation Of Parallel Performancementioning

confidence: 71%

“…Using the terminology of [8], let R = Z[x, x −1 ] be the ring of Laurent polynomials in an indeterminate x. For the purpose of this paper, it suffices to know that a Hecke algebra 5 H is an R-algebra with a basis {T w | w ∈ W } over R, where W is a finite Coxeter group with set of generators S. In this paper, we only consider Hecke algebras of type E m (m = 6,7,8), that is, W is the exceptional Coxeter group E m , and the cardinality of the set of generators S is m.…”

Section: Algorithm For Finding Invariant Bilinear Formsmentioning

confidence: 99%

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High-Performance Computer Algebra: A Hecke Algebra Case Study

Maier

Livesey

Loidl³

et al. 2014

Lecture Notes in Computer Science

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Abstract. We describe the first ever parallelisation of an algebraic computation at modern HPC scale. Our case study poses challenges typical of the domain: it is a multi-phase application with dynamic task creation and irregular parallelism over complex control and data structures. Our starting point is a sequential algorithm for finding invariant bilinear forms in the representation theory of Hecke algebras, implemented in the GAP computational group theory system. After optimising the sequential code we develop a parallel algorithm that exploits the new skeleton-based SGP2 framework to parallelise the three most computationally-intensive phases. To this end we develop a new domain-specific skeleton, parBufferTryReduce. We report good parallel performance both on a commodity cluster and on a national HPC, delivering speedups up to 548 over the optimised sequential implementation on 1024 cores.

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“…In the following, we will only sketch this case study and present the main per- formance results; the reader is referred to [6] for a detailed report and to [62] for background on the problem. An n-dimensional representation of H is an R-algebra homomorphism from H to M n .R/, the R-algebra of n n matrices over R D ZOEx; x 1 , the ring of Laurent polynomials in indeterminate x.…”

Section: Hecke Algebras In Sgp2mentioning

confidence: 99%

HPC‐GAP: engineering a 21st‐century high‐performance computer algebra system

Behrends

Hammond

Janjić

et al. 2016

Concurrency and Computation

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SUMMARYSymbolic computation has underpinned a number of key advances in Mathematics and Computer Science. Applications are typically large and potentially highly parallel, making them good candidates for parallel execution at a variety of scales from multi-core to high-performance computing systems. However, much existing work on parallel computing is based around numeric rather than symbolic computations. In particular, symbolic computing presents particular problems in terms of varying granularity and irregular task sizes that do not match conventional approaches to parallelisation. It also presents problems in terms of the structure of the algorithms and data. This paper describes a new implementation of the free open-source GAP computational algebra system that places parallelism at the heart of the design, dealing with the key scalability and cross-platform portability problems. We provide three system layers that deal with the three most important classes of hardware: individual shared memory multi-core nodes, mid-scale distributed clusters of (multi-core) nodes and full-blown high-performance computing systems, comprising large-scale tightly connected networks of multi-core nodes. This requires us to develop new cross-layer programming abstractions in the form of new domain-specific skeletons that allow us to seamlessly target different hardware levels. Our results show that, using our approach, we can achieve good scalability and speedups for two realistic exemplars, on high-performance systems comprising up to 32 000 cores, as well as on ubiquitous multi-core systems and distributed clusters. The work reported here paves the way towards full-scale exploitation of symbolic computation by high-performance computing systems, and we demonstrate the potential with two major case studies.

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Hecke algebras of finite type are cellular

Cited by 76 publications

References 24 publications

Quiver Varieties, Category for Rational Cherednik Algebras, and Hecke Algebras

Quiver Varieties, Category for Rational Cherednik Algebras, and Hecke Algebras

High-Performance Computer Algebra: A Hecke Algebra Case Study

HPC‐GAP: engineering a 21st‐century high‐performance computer algebra system

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