Representations of Khovanov–Lauda–Rouquier algebras and combinatorics of Lyndon words

By Alexander S. Kleshchev, Arun Ram. 2009

99 citations recorded by the source corpus.

We construct irreducible representations of affine Khovanov–Lauda–Rouquier algebras of arbitrary finite type. The irreducible representations arise as simple heads of appropriate induced modules, and thus our construction is similar to that of Bernstein and Zelevinsky for affine Hecke algebras of type A. The highest weights of irreducible modules are given by the so-called good words, and the highest weights of the ‘cuspidal modules’ are given by the good Lyndon words. In a sense, this has been predicted by Leclerc.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3e97db870a15c4b65aaa318f92a8c3a839750c78

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In Computational Theory and Mathematics

1. Ben Webster
2. Peter J. McNamara
3. Aaron D. Lauda
4. Sabin Cautis
5. Monica Vazirani
6. E. Witten

In Discrete Mathematics and Combinatorics

1. Aaron D. Lauda
2. Sabin Cautis
3. Peter J. McNamara
4. Ben Webster
5. Monica Vazirani
6. Mikhail Khovanov
7. Raphaël Rouquier

In Geometry and Topology

1. Aaron D. Lauda
2. Ben Webster
3. Sabin Cautis
4. Mikhail Khovanov
5. Monica Vazirani
6. David E. V. Rose
7. Peter J. McNamara
8. Hoel Queffelec

Citations in the corpus

• Implicit structure in 2-representations of quantum groups (2011, background): “Z is the integral version of the irreducible highest weight representation for U(˙ g) associated to Λ and [Proj( RΛ)] is the Grothendieck ring of Proj( ). Parts of this conjecture have been proven in [4, 19, 3, 24, 20, 14]. Recently Kang and Kashiwara [13] prove this conjecture by showing that i-ind”
• Knot invariants and higher representation theory I: diagrammatic and geometric categorification of tensor products (2010, background): “ ideal generated by the elementsyλi1 1 e(i). We letVλdenote the category of finite dimensionalRλ-modules. ThisalgebrahasattractedgreatinterestrecentlyintheworkofBrundan-Kleshchev [BK09], Kleshchev-Ram [KR], Hoffnung-Lauda and Lauda-Vazirani [LV, HL], and Hill-Melvin-Mondragon [HMM]. It has a very rich”
• Finite Dimensional Representations of Khovanov-Lauda-Rouquier algebras I: Finite Type (2012, background): “e representations of these algebras. There are various answers to this question dating back to [KL1]. Let us focus our attention on the classification in terms of Lyndon words due to Kleshchev and Ram [KR] and Hill, Melvin and Mondragon [HMM], where a family of standard modules are constructed from w”
• Crystals from categorified quantum groups (2009, methodology): “In fact, Kleshchev and Ram have conjectured [KR09] that the dimensions of all simple R(ν)-modules are independent of chark for finite Cartan datum.”