Admissible diagrams in quantum nilpotent algebras and combinatoric properties of Weyl groups
Admissible diagrams in quantum nilpotent algebras and combinatoric properties of Weyl groups
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量子幂零代数中的容许图和 Weyl 群的组合性质
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发表时间：
2010
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通讯作者：
G. Cauchon
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作者：
Antoine Mériaux;G. Cauchon
考虑一个秩为n的复单李代数g。用W表示单根系，用W表示相应的Weyl群，考虑某个w ∈ W的约化表达式w = sα1 <$··<$sαt（每个αi ∈ <$），调用图1，. . .，t .我们用U+[w]（或Uw q（g））表示Jantzen在1996年定义的“量子幂零”代数。我们证明（定理5.3.1）：在R意义下，正图自然地与（上面选择的w的约化表达式的）正子表达式相关联。Marsh和K. Rietsch（或等价地，在V. Deodhar的意义上没有缺陷的子表达式），与G. Cauchon描述了Spec（U+[w]）的自然分层。这个定理特别地暗示（推论5.3.1和5.3.2）：（1）映射ε：Δ = {j1 < · · · < js} → u = sαj1 ε· · ·ε sαjs是从容许图集合到集合{u ∈ W| u ≤ w}。(2)对于每个容许图Δ = {j1 < · · · < js}，sαj1 <$···<$sαjs是u =<$（Δ）的约化表达式. (3)映射ε ′：Δ = {j1 < · · · < js} → u′ = sαjs ε· · ·ε sαj1是从容许图集到集合{u ∈ W}的双射|u ≤ v = w−1}。(4)对于每个容许图Δ = {j1 < · · · < js}，sαjs <$···<$sαj1是u′ =<$′（Δ）的约化表达式.如果李代数g是An型的，并且选择w使得U+[w]是量子矩阵Oq（Mp，m（k））的代数，其中m = n-p + 1（见2.1节），则容许图是A意义下的Γ -图。Postnikov（http：//arxiv.org/abs/math/0609764）。在这个特殊的例子中，断言3和4也被A证明了（用完全不同的方法）。Postnikov和T. Lam和L.威廉姆斯。
Consider a complex simple Lie algebra g of rank n. Denote by Π a system of simple roots, by W the corresponding Weyl group, consider a reduced expression w = sα1 ◦ · · · ◦ sαt (each αi ∈ Π) of some w ∈ W and call diagram any subset of 1, . . . , t . We denote by U+[w] (or Uw q (g)) the “quantum nilpotent” algebra as defined by Jantzen in 1996 We prove (theorem 5.3.1) that the positive diagrams naturally associated with the positive subexpressions (of the reduced expression of w chosen above) in the sense of R. Marsh and K. Rietsch (or equivalently the subexpressions without defect in the sense of V. Deodhar), coincide with the admissible diagrams constructed by G. Cauchon which describe the natural stratification of Spec(U+[w]). This theorem implies in particular (corollaries 5.3.1 and 5.3.2): (1) The map ζ : Δ = {j1 < · · · < js} → u = sαj1 ◦ · · · ◦ sαjs is a bijection from the set of admissible diagrams onto the set {u ∈ W |u ≤ w}. (2) For each admissible diagram Δ = {j1 < · · · < js}, sαj1 ◦ · · · ◦ sαjs is a reduced expression of u = ζ(Δ). (3) The map ζ′ : Δ = {j1 < · · · < js} → u′ = sαjs ◦ · · · ◦ sαj1 is a bijection from the set of admissible diagrams onto the set {u ∈ W |u ≤ v = w−1}. (4) For each admissible diagram Δ = {j1 < · · · < js}, sαjs ◦ · · · ◦ sαj1 is a reduced expression of u′ = ζ′(Δ). If the Lie algebra g is of type An and w is chosen in order that U+[w] is the algebra of quantum matrices Oq(Mp,m(k)) with m = n − p + 1 (see section 2.1), then, the admissible diagrams are the Γ -diagrams in the sense of A. Postnikov (http://arxiv.org/abs/math/0609764). In this particular case, the assertions 3 and 4 have also been proved (with quite different methods) by A. Postnikov and by T. Lam and L. Williams.