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## A Steinberg type decomposition theorem for higher level Demazure modules (2014)

Citations: | 3 - 1 self |

### Citations

1048 |
Infinite-dimensional Lie algebras
- Kac
- 1990
(Show Context)
Citation Context ...V : h ∗ → Z+, by sending µ→ dimVµ. If wtV is a finite set, then chh V = ∑ µ∈h∗ dimVµe(µ) ∈ Z[P ]. 2.4. We now define the untwisted affine Lie algebra associated to g and some related terminology (see =-=[25]-=- for details). The affine Lie algebra ĝ is given by ĝ = g⊗ C[t, t−1]⊕ Cc⊕ Cd where c is the canonical central element, and d acts as the derivation t d dt and commutator [x⊗ tr, y ⊗ ts] = [x, y]⊗ tr... |

265 |
Kac–Moody groups, their flag varieties and representation theory
- Kumar
- 2002
(Show Context)
Citation Context ...0) of Ŵ and s0 = (sθ, tθ). Given a reduced expression w = si1 · · · sir for an element w ∈ Ŵ , set Dw = Di1 · · ·Dir , and note that Dw is independent of the choice of reduced expression for w (see =-=[28]-=-, Corollary 8.2.10). For σ ∈ Σ, and w ∈ Ŵ , set Dwσ(e(Λ)) = Dw(e(σ(Λ)). Since Di(e(δ)) = e(δ), it follows that for all w ∈ W̃ , the operator Dw descends to Z[P̂ ]/Iδ . The following result is proved ... |

87 | Weyl modules for classical and quantum affine algebras
- Chari, A
(Show Context)
Citation Context ...r slr+1 and in [23] in general. Our interest in these modules arise from their relationship with the representation theory of quantum affine algebras. This connection was originally developed in [4], =-=[10]-=-, [12] where it was shown that the classical limit of certain irreducible representations of the quantum affine algebra can be viewed as graded representations of g[t]. The classical limits were first... |

72 | On generalized Kostka polynomials and quantum Verlinde rule
- Feigin, Loktev
(Show Context)
Citation Context ...n that paper, the connection was also made between these modules and the V.C. was partially supported by DMS-0901253 and DMS- 1303052. 1 2 CHARI, SHEREEN, VENKATESH AND WAND fusion product defined in =-=[16]-=- of representations of g[t]. In [12] it was shown that a Kirillov– Reshetikhin module for a quantum affine algebra is similarly related to a Demazure module when g is of classical type. In [17] and [1... |

71 | Cluster algebras and quantum affine algebras
- Hernandez, Leclerc
(Show Context)
Citation Context ...to study this question is the following: when ℓ = 2 and in the case of sln+1, these modules are related to the modules for the quantum affine algebra which occur in the work of Hernandez–Leclerc (see =-=[22]-=-). This relationship is made precsie in [1]. Recall that Steinberg’s tensor product theorem asserts that a simple module L(λ) of an algebraic group over characteristic p is isomorphic to a tensor prod... |

64 | Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions, Adv
- Fourier, Littelmann
(Show Context)
Citation Context ...6] of representations of g[t]. In [12] it was shown that a Kirillov– Reshetikhin module for a quantum affine algebra is similarly related to a Demazure module when g is of classical type. In [17] and =-=[18]-=- the authors worked with arbitrary untwisted affine Lie algebras and with particular classes of g[t]–stable Demazure module . In the simply–laced case for instance, they studied the modules D(ℓ, ℓµ) w... |

62 | On the fermionic formula and the Kirillov-Reshetikhin conjecture
- Chari
(Show Context)
Citation Context ...6] for slr+1 and in [23] in general. Our interest in these modules arise from their relationship with the representation theory of quantum affine algebras. This connection was originally developed in =-=[4]-=-, [10], [12] where it was shown that the classical limit of certain irreducible representations of the quantum affine algebra can be viewed as graded representations of g[t]. The classical limits were... |

62 | t-analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras, Represent. Theory 7
- Nakajima
- 2003
(Show Context)
Citation Context ...hat any Demazure module is a fusion product of prime Demazure modules. We use our main result to study generalizations of Q–systems (see [20] for details, [27] for a more recent discussion and [21] , =-=[32]-=- for the quantum analog). In the case of sln+1, the Q–system is a classical identity of Schur functions associated to rectangular weights of a fixed height. Equivalently, the Q–system is a short exact... |

52 |
Remarks on fermionic formula. In Recent developments in quantum affine algebras and related topics
- Hatayama, Kuniba, et al.
- 1998
(Show Context)
Citation Context ... g–modules. In the case when g is of type type A or D we show that any Demazure module is a fusion product of prime Demazure modules. We use our main result to study generalizations of Q–systems (see =-=[20]-=- for details, [27] for a more recent discussion and [21] , [32] for the quantum analog). In the case of sln+1, the Q–system is a classical identity of Schur functions associated to rectangular weights... |

51 |
Demazure character formula in arbitrary Kac-Moody setting
- Kumar
- 1987
(Show Context)
Citation Context ...m 1. As in [17] and [37], the proof uses the Demazure operators and the Demazure character formula in a crucial way. We recollect these concepts briefly and refer the interested reader to [14], [17], =-=[29]-=- and [31] for a more detailed discussion. 4.1. There are two main ingredients in the proof of the Theorem. The first is the following proposition which was proved in [37] but we include a very brief s... |

50 |
The Kirillov-Reshetikhin conjecture and solutions of T -systems
- Hernandez
(Show Context)
Citation Context ... show that any Demazure module is a fusion product of prime Demazure modules. We use our main result to study generalizations of Q–systems (see [20] for details, [27] for a more recent discussion and =-=[21]-=- , [32] for the quantum analog). In the case of sln+1, the Q–system is a classical identity of Schur functions associated to rectangular weights of a fixed height. Equivalently, the Q–system is a shor... |

44 |
Formules de caracteres pour les algebres de Kac-Moody generales,
- Mathieu
- 1988
(Show Context)
Citation Context ...n [17] and [37], the proof uses the Demazure operators and the Demazure character formula in a crucial way. We recollect these concepts briefly and refer the interested reader to [14], [17], [29] and =-=[31]-=- for a more detailed discussion. 4.1. There are two main ingredients in the proof of the Theorem. The first is the following proposition which was proved in [37] but we include a very brief sketch of ... |

39 |
Q-Systems as cluster algebras
- Kedem
- 2008
(Show Context)
Citation Context ... case when g is of type type A or D we show that any Demazure module is a fusion product of prime Demazure modules. We use our main result to study generalizations of Q–systems (see [20] for details, =-=[27]-=- for a more recent discussion and [21] , [32] for the quantum analog). In the case of sln+1, the Q–system is a classical identity of Schur functions associated to rectangular weights of a fixed height... |

37 | Tensor product structure of affine Demazure modules and limit constructions
- Fourier, Littelmann
(Show Context)
Citation Context ...ned in [16] of representations of g[t]. In [12] it was shown that a Kirillov– Reshetikhin module for a quantum affine algebra is similarly related to a Demazure module when g is of classical type. In =-=[17]-=- and [18] the authors worked with arbitrary untwisted affine Lie algebras and with particular classes of g[t]–stable Demazure module . In the simply–laced case for instance, they studied the modules D... |

36 | Lie algebras of finite and affine type - Carter - 2005 |

36 |
Demazure and fusion modules for the current algebra of slr+1,
- Chari, Loktev, et al.
- 2006
(Show Context)
Citation Context ...ntum affine algebra can be viewed as graded representations of g[t]. The classical limits were first related to the g[t]–stable Demazure modules in level one representations of affine Lie algebras in =-=[8]-=- for slr+1. In that paper, the connection was also made between these modules and the V.C. was partially supported by DMS-0901253 and DMS- 1303052. 1 2 CHARI, SHEREEN, VENKATESH AND WAND fusion produc... |

35 | Nonsymmetric Macdonald polynomials and Demazure characters
- Ion
- 2003
(Show Context)
Citation Context ...rresponding module by D(ℓ, λ). In the case when ℓ = 1, these modules are interesting for a variety of reasons, including the connection with Macdonald polynomials established in [36] for slr+1 and in =-=[23]-=- in general. Our interest in these modules arise from their relationship with the representation theory of quantum affine algebras. This connection was originally developed in [4], [10], [12] where it... |

33 | Beyond Kirillov-Reshetikhin modules, Quantum affine algebras, extended affine Lie algebras, and their applications
- Chari, Hernandez
- 2010
(Show Context)
Citation Context ... [4],[21], [32], [26]) together with [12] shows that the g[t]–module D(ℓ, ℓωi) is the “limit”of the corresponding Kirillov–Reshetikhin modules. Other examples of prime representations can be found in =-=[7]-=-, A STEINBERG TYPE DECOMPOSITION THEOREM FOR HIGHER LEVEL DEMAZURE MODULES 11 [12], [22]. In all these examples one actually proves that the underlying g–module is prime which motivates the following ... |

33 | The restricted Kirillov-Reshetikhin modules for the current and twisted current algebras
- Chari, Moura
(Show Context)
Citation Context ...1 and in [23] in general. Our interest in these modules arise from their relationship with the representation theory of quantum affine algebras. This connection was originally developed in [4], [10], =-=[12]-=- where it was shown that the classical limit of certain irreducible representations of the quantum affine algebra can be viewed as graded representations of g[t]. The classical limits were first relat... |

32 | Weyl modules, Demazure modules and finite crystals for nonsimply laced type - Naoi |

30 |
Une nouvelle formule des caractères
- Demazure
- 1974
(Show Context)
Citation Context ...dules for a simple Lie algebra. 1. Introduction Demazure modules associated to simple Lie algebra or more generally a Kac–Moody Lie algbera g have been studied intensively since their introduction in =-=[14]-=-. These modules, which are actually modules for a Borel subalgebra of the Lie algebra, are indexed by a dominant integral weight Λ and an element w of the Weyl group. In this paper we shall be concern... |

27 | A categorical approach to Weyl modules - Chari, Fourier, et al. |

26 |
On the connection between Macdonald polynomials and Demazure characters
- Sanderson
(Show Context)
Citation Context ...g and we denote the corresponding module by D(ℓ, λ). In the case when ℓ = 1, these modules are interesting for a variety of reasons, including the connection with Macdonald polynomials established in =-=[36]-=- for slr+1 and in [23] in general. Our interest in these modules arise from their relationship with the representation theory of quantum affine algebras. This connection was originally developed in [4... |

20 | Variétés de Schubert et excellentes filtrations. Astérisque - Polo - 1989 |

16 | On the Demazure character formula - Joseph - 1985 |

10 | Construction du groupe de Kac-Moody et applications - Mathieu - 1988 |

10 | Unique decomposition of tensor products of irreducible representations of simple algebraic groups.
- Rajan
- 2004
(Show Context)
Citation Context ...ensor product of non–trivial prime modules. However, it is not known in general if such a decomposition is unique. The uniqueness of a tensor product of simple g–modules was proved fairly recently in =-=[35]-=-, [38]. Notice that a g[t]–module V which is prime is necessarily prime with respect to the fusion product as well. 3.9. Our final result shows that if g is of type A or D, then any Demazure module is... |

9 | Demazure modules, fusion products and Q–systems
- Chari, Venkatesh
(Show Context)
Citation Context ...n particular, the fusion product on the right hand side is independent of the choice of parameters. 3.5. In the case when λ0 = 0 the result was first proved in [18] and a different proof was given in =-=[11]-=-. As in these papers, the proof of our theorem uses the theory of Demazure operators and the following additional key result proved in Section 7. Proposition. Assume that g is of classical type or of ... |

8 |
Factorization of representations of quantum affine algebras, Modular interfaces
- CHARI, PRESSLEY
- 1997
(Show Context)
Citation Context ...and interesting to talk about the prime irreducible representations: namely an irreducible representation which is not isomorphic to the tensor product of non–trivial irreducible representations (see =-=[9]-=-, [13], [22]). An important family of prime irreducible representations are the Kirillov–Reshetikhin modules. Using the work of several authors ([10], [4],[21], [32], [26]) together with [12] shows th... |

6 |
q–characters of the tensor products in sl2–case, Mosc
- Feigin, Feigin
(Show Context)
Citation Context ...· · · ∗ vm of this element. Clearly the definiton of the fusion product depends on the parameters zs, 1 ≤ s ≤ k. However it is conjectured in [16] and (proved in certain cases by various people, [8], =-=[15]-=-, [16] [18], [26] for instance) that under suitable conditions on Vs and vs, the fusion product is independent of the choice of the complex numbers. For ease of notation we shall often suppress the de... |

6 |
A pentagon of identities, graded tensor products, and the Kirillov-Reshetikhin conjecture
- Kedem
- 2011
(Show Context)
Citation Context ...s element. Clearly the definiton of the fusion product depends on the parameters zs, 1 ≤ s ≤ k. However it is conjectured in [16] and (proved in certain cases by various people, [8], [15], [16] [18], =-=[26]-=- for instance) that under suitable conditions on Vs and vs, the fusion product is independent of the choice of the complex numbers. For ease of notation we shall often suppress the dependence on the c... |

4 | Fusion product structure of Demazure modules. arXiv:1311.2224
- Venkatesh
- 2013
(Show Context)
Citation Context ... these prime Demazure modules to prime representations of quantum affine algebras is studied. 4. Proof of Theorem 1 In this section we shall assume Proposition 3.5 and prove Theorem 1. As in [17] and =-=[37]-=-, the proof uses the Demazure operators and the Demazure character formula in a crucial way. We recollect these concepts briefly and refer the interested reader to [14], [17], [29] and [31] for a more... |

3 | Prime representations from a homological perspective. arXiv:1112.6376
- Chari, Moura, et al.
- 1994
(Show Context)
Citation Context ...nteresting to talk about the prime irreducible representations: namely an irreducible representation which is not isomorphic to the tensor product of non–trivial irreducible representations (see [9], =-=[13]-=-, [22]). An important family of prime irreducible representations are the Kirillov–Reshetikhin modules. Using the work of several authors ([10], [4],[21], [32], [26]) together with [12] shows that the... |

2 | Lie Groups and Lie Algebras IV-VI - Bourbaki - 2000 |

2 |
Unique factorization of tensor products for KacMoody algebras
- Venkatesh, Viswanath
(Show Context)
Citation Context ...product of non–trivial prime modules. However, it is not known in general if such a decomposition is unique. The uniqueness of a tensor product of simple g–modules was proved fairly recently in [35], =-=[38]-=-. Notice that a g[t]–module V which is prime is necessarily prime with respect to the fusion product as well. 3.9. Our final result shows that if g is of type A or D, then any Demazure module is a fus... |

1 |
Prime representations and Demazure modules, in preparation
- Brito, Chari, et al.
(Show Context)
Citation Context ...n ℓ = 2 and in the case of sln+1, these modules are related to the modules for the quantum affine algebra which occur in the work of Hernandez–Leclerc (see [22]). This relationship is made precsie in =-=[1]-=-. Recall that Steinberg’s tensor product theorem asserts that a simple module L(λ) of an algebraic group over characteristic p is isomorphic to a tensor product L(pλ1)⊗ L(λ0) where λ0(hi) ≤ p for all ... |

1 | Schur positivity and Kirillov–Reshetikhin modules - Fourier, Hernandez - 2014 |