### Citations

665 |
Reflection groups and Coxeter groups, Cambridge
- Humphreys
- 1990
(Show Context)
Citation Context ...elow. W h Φ ′ = W.α0 Wα0 type f Φ′ (q)/[h]q gcd([h]q, ∑ i qd∗i ) An−1 n Φ An−3 [n − 1]q 1 Bn 2n ± ej} A1 × Bn−2 [n − 1] q2 [n] q2 {±ei} Bn−1 1 Dn 2(n − 1) Φ A1 × Dn−2 [n−2] q 2 [n]q E6 12 Φ A5 [2] q 4=-=[3]-=- q 3 1 E7 18 Φ D6 [2]q 1 [2] q6 [2] q2 [7] q2 1 E8 30 Φ E7 [2] q 10[4] q 6 1 F4 12 either orbit B3 [2] q 4 [2] q 6 H3 10 Φ A1 × A1 [3] q2 1 H4 30 Φ H3 [2] q6[2] q10 1 I2(m) m either orbit A1 1 1 if m ... |

297 |
Finite unitary reflection groups
- Shephard, Todd
- 1954
(Show Context)
Citation Context ...ents. 1. Introduction Consider a complex reflection group W ⊂ GL(V ) with V = Cℓ , acting by linear substitutions on the polynomial algebra S = Sym(V ∗ ) ∼ = C[x1, . . . , xn]. Both Shephard and Todd =-=[9]-=- and Chevalley [2] proved that the invariant subalgebra is again a polynomial algebra SW = C[f1, . . . , fℓ] for some homogeneous polynomials fi, and that the coinvariant algebra S/I where I = (f1, . ... |

172 | Geometric Langlands duality and representations of algebraic groups over commutative rings
- Mirković, Vilonen
- 2007
(Show Context)
Citation Context ...ty G/P also arises as a Schubert variety in the affine Grassmannian. The cell decomposition of G/P as above can be used to give a decomposition of this cone into Mirković-Vilonen cycles introduced in =-=[5]-=-. In this picture, the dimension formula for the Mirković-Vilonen cycles is equivalent to Lemma 4; see Mirković and Vilonen [5, Theorem 3.2] with λ = α0, and also Ngô and Polo [6, Lemme 7.4]. 4.4. A-D... |

169 | Lie groups and Lie algebras, Chapters 4-6 - Bourbaki - 2008 |

163 |
Invariants of finite groups generated by reflections
- Chevalley
- 1955
(Show Context)
Citation Context ...ion Consider a complex reflection group W ⊂ GL(V ) with V = Cℓ , acting by linear substitutions on the polynomial algebra S = Sym(V ∗ ) ∼ = C[x1, . . . , xn]. Both Shephard and Todd [9] and Chevalley =-=[2]-=- proved that the invariant subalgebra is again a polynomial algebra SW = C[f1, . . . , fℓ] for some homogeneous polynomials fi, and that the coinvariant algebra S/I where I = (f1, . . . , fℓ) carries ... |

146 | groups and Lie algebras. Chapters 4–6. Elements of Mathematics - Lie - 2002 |

130 |
Regelar elements of finite reflection groups
- Springer
- 1974
(Show Context)
Citation Context ...W acts transitively on the subset Φ ′ of Φ. The desired divisibility will then be deduced from Lemma 2 below, applied to a Coxeter element of W . The statement of the lemma involves Springer’s notion =-=[11]-=- of a regular element c in W , with a regular eigenvalue ζ, meaning c(v) = ζv for an eigenvector v lying on none of the reflecting hyperplanes for W . Then c and ζ have the same multiplicative order n... |

79 | The cyclic sieving phenomenon,
- Reiner, Stanton, et al.
- 2004
(Show Context)
Citation Context ...tion U = CX satisfies f X (ζ m ) = #{x ∈ X : c m (x) = x}. In particular, f X (q) is divisible by [n]q if and only if C acts freely on X. Proof. For the sake of completeness, we recall the proof from =-=[8]-=-. Springer [11] extended the work of Shephard-Todd and Chevalley by proving one has an isomorphism W × C-representations (2.1) S/I ∼ = CW where W acts as before, and where C acts on CW via right-trans... |

53 |
Invariants of finite reflection groups
- Solomon
- 1963
(Show Context)
Citation Context ...ction group, Weyl group, fake degree, codegree, simply-laced. First, second authors partially supported by the NSF grants DMS-0601010, DMS-0969470. 1 This follows as a consequence of Solomon’s result =-=[10]-=- that the W -invariant differential forms with polynomial coefficients S ⊗ ∧ k V form a free S W -module with basis elements dfi1 ∧ · · · ∧ dfi k . 1 ℓ ). V ∗2 VICTOR REINER AND ZHIWEI YUN 2. Proof o... |

28 | Résolutions de Demazure affines et formule de Casselman-Shalika géométrique - Ngô, Polo |

7 | Coxeter elements and periodic Auslander-Reiten quiver
- Kirillov, Thind
(Show Context)
Citation Context ...n Φ An−3 [n − 1]q 1 Bn 2n ± ej} A1 × Bn−2 [n − 1] q2 [n] q2 {±ei} Bn−1 1 Dn 2(n − 1) Φ A1 × Dn−2 [n−2] q 2 [n]q E6 12 Φ A5 [2] q 4[3] q 3 1 E7 18 Φ D6 [2]q 1 [2] q6 [2] q2 [7] q2 1 E8 30 Φ E7 [2] q 10=-=[4]-=- q 6 1 F4 12 either orbit B3 [2] q 4 [2] q 6 H3 10 Φ A1 × A1 [3] q2 1 H4 30 Φ H3 [2] q6[2] q10 1 I2(m) m either orbit A1 1 1 if m 2 odd m even [2] q2 if m 2 even I2(m) m Φ − [2]q 1 m odd The table exh... |

7 | Quasi-minuscule quotients and reduced words for reflections - Stembridge |

6 | Spectra of symmetrized shuffling operators
- Reiner, Saliola, et al.
(Show Context)
Citation Context ...q gcd([h]q, ∑ i qd∗i ) An−1 n Φ An−3 [n − 1]q 1 Bn 2n ± ej} A1 × Bn−2 [n − 1] q2 [n] q2 {±ei} Bn−1 1 Dn 2(n − 1) Φ A1 × Dn−2 [n−2] q 2 [n]q E6 12 Φ A5 [2] q 4[3] q 3 1 E7 18 Φ D6 [2]q 1 [2] q6 [2] q2 =-=[7]-=- q2 1 E8 30 Φ E7 [2] q 10[4] q 6 1 F4 12 either orbit B3 [2] q 4 [2] q 6 H3 10 Φ A1 × A1 [3] q2 1 H4 30 Φ H3 [2] q6[2] q10 1 I2(m) m either orbit A1 1 1 if m 2 odd m even [2] q2 if m 2 even I2(m) m Φ ... |

4 | Graded multiplicities in the Macdonald kernel - Stembridge |