BGG resolution - 편집 역사

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imported>Pythagoras0: section 'articles' updated

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section 'articles' updated

imported>Pythagoras0: /* introduction */

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introduction

imported>Pythagoras0: /* introduction */

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introduction

imported>Pythagoras0: /* articles */

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articles

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← 이전 판	2020년 11월 16일 (월) 12:25 판
1번째 줄:	1번째 줄:
	+	==introduction==
	+	* <math>L(\cdot)</math> : simple module, <math>V(\cdot)</math> : Verma module
	+	* Weyl character formula. For <math>\lambda\in \Lambda^{+}</math>,
	+	:<math>
	+	\operatorname{ch}L(\lambda)=\sum_{w \in W}(-1)^{\ell(w)}\operatorname{ch} M(w\cdot \lambda)
	+	</math>
	+	* goal : realize this formula as an Euler characteristic
	+	* The BGG resolution resolves a finite-dimensional simple <math>\mathfrak{g}</math>-module <math>L(\lambda)</math> by direct sums of Verma modules indexed by weights "of the same length" in the orbit <math>W\cdot \lambda</math>
	+	;thm (Bernstein-Gelfand-Gelfand Resolution).
	+	Let <math>\lambda\in \Lambda^{+}</math>. There is an exact sequence of Verma modules
	+	:<math>
	+	0 \to M({w_0\cdot \lambda})\to \cdots \to \bigoplus_{w\in W, \ell(w)=k}M({w\cdot \lambda})\to \cdots \to M({\lambda})\to L({\lambda})\to 0
	+	</math>
	+	where <math>\ell(w)</math> is the length of the Weyl group element <math>w</math>, <math>w_0</math> is the Weyl group element
	+	of maximal length. Here <math>\rho</math> is half the sum of the positive roots.

	+	==applications==
	+	* This is used to compute the cohomologies of <math>\mathfrak{n}^+</math>.
	+	* see [[Kostant theorem on Lie algebra cohomology of nilpotent subalgebra]]
	+
	+
	+	==generalization of BGG resolution==
	+	* There exist generalizations to symmetrizable Kac-Moody algebras, cf. [34].
	+	* Kempf obtained a resolution of finite-dimensional L(λ) in terms of the Grothendieck-Cousin complex in [26], which is dual to the BGG resolution.
	+	* This was extended by Kumar to arbitrary Kac-Moody algebras; he thus obtained the BGG resolution here, and computed the Weyl-Kac character formula and the cohomologies of n+ (cf. [30,§9.3]).
	+
	+	==related items==
	+	* [[Talk on BGG resolution]]
	+	* [[Verma modules]]
	+	* [[BGG reciprocity]]
	+	* [[BGG category]]
	+	* [[Kostant theorem on Lie algebra cohomology of nilpotent radical]]
	+	* [[Bott-Borel-Weil Theorem]]
	+	* [[Koszul complex]]
	+
	+	==books==
	+	* [30] Shrawan Kumar, Kac-Moody Groups, their Flag Varieties and Representation Theory, Birkhauser, Progress in Math. 204, Boston, 2002
	+	* James E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category O, Grad. Stud. Math., 94, Amer. Math. Soc., Providence, RI, 2008.
	+
	+
	+	==expositions==
	+	* http://rvirk.com/notes/student/catObasics.pdf
	+	* BGG resolution http://www.math.columbia.edu/~woit/LieGroups-2012/vermamodules.pdf
	+	* Wang, Jing Ping. “Representations of sl(2,C) in the BGG Category O and Master Symmetries.” arXiv:1408.3437 [nlin], August 14, 2014. http://arxiv.org/abs/1408.3437.
	+	* http://stanford.edu/~khare/EoM-BGG-O.pdf
	+
	+	==articles==
	+	* Pierre Julg, The Bernstein-Gelfand-Gelfand complex for rank one semi simple Lie groups as a Kasparov module, arXiv:1605.07408 [math.OA], May 24 2016, http://arxiv.org/abs/1605.07408
	+	* Griffeth, Stephen, and Emily Norton. “Character Formulas and Bernstein-Gelfand-Gelfand Resolutions for Cherednik Algebra Modules.” arXiv:1511.00748 [math], November 2, 2015. http://arxiv.org/abs/1511.00748.
	+	* Zelevinski, Resolvents, dual pairs, and character formulas http://www.ms.unimelb.edu.au/~ram/Resources/ResolventsDualPairsAndCharacterFormulas.html
	+	* [34] A. Rocha-Caridi, Splitting Criteria for <math>\mathfrak{g}</math>-modules induced from a parabolic and the Bernstein-Gelfand-Gelfand resolution of a finite-dimensional, irreducible <math>\mathfrak{g}</math>-module, Trans. Amer. Math. Soc.262 (1980), no. 2, 335–366
	+	* [26] G. Kempf, The Grothendieck-Cousin complex of an induced representation , Advances in Mathematics 29 (1978), 310–396
	+	* [31] Lepowsky, J. “A Generalization of the Bernstein-Gelfand-Gelfand Resolution.” Journal of Algebra 49, no. 2 (1977): 496–511.
	+	* J. Bernstein, I. Gel'fand, and S. Gel'fand, A category of g-modules, Functional Anal. Appl. 10 (1976), 87-92
	+	* [5] Bernšteĭn, I. N., I. M. Gel'fand, and S. I. Gel'fand. ‘Differential Operators on the Base Affine Space and a Study of <math>\mathfrak{g}</math>-Modules’. In Lie Groups and Their Representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), 21–64. Halsted, New York, 1975. http://www.ams.org/mathscinet-getitem?mr=0578996.
	+	* Bernšteĭn, I. N., I. M. Gel'fand, and S. I. Gel'fand. ‘Structure of Representations That Are Generated by Vectors of Highest Weight’. Akademija Nauk SSSR. Funkcional\cprime Nyi Analiz I Ego Priloženija 5, no. 1 (1971): 1–9.
	+
	+	[[분류:Lie theory]]
	+	[[분류:abstract concepts]]
	+	[[분류:migrate]]

← 이전 판		2020년 11월 12일 (목) 11:52 판
1번째 줄:		1번째 줄:
−	~~==introduction==~~
−	* $L(\cdot)$ : simple module, $V(\cdot)$ : Verma module
−	* Weyl character formula. For $\lambda\in \Lambda^{+}$,
−	$$
−	~~\operatorname{ch}L(\lambda)=\sum_{w \in W}(-1)^{\ell(w)}\operatorname{ch} M(w\cdot \lambda)~~
−	$$
−	* goal : realize this formula as an Euler characteristic
−	* The BGG resolution resolves a finite-dimensional simple $\mathfrak{g}$-module $L(\lambda)$ by direct sums of Verma modules indexed by weights "of the same length" in the orbit $W\cdot \lambda$
−	~~;thm (Bernstein-Gelfand-Gelfand Resolution).~~
−	~~Let $\lambda\in \Lambda^{+}$. There is an exact sequence of Verma modules~~
−	$$
−	~~0 \to M({w_0\cdot \lambda})\to \cdots \to \bigoplus_{w\in W, \ell(w)=k}M({w\cdot \lambda})\to \cdots \to M({\lambda})\to L({\lambda})\to 0~~
−	$$
−	~~where $\ell(w)$ is the length of the Weyl group element $w$, $w_0$ is the Weyl group element~~
−	~~of maximal length. Here $\rho$ is half the sum of the positive roots.~~

−	~~==applications==~~
−	* This is used to compute the cohomologies of $\mathfrak{n}^+$.
−	* see [[Kostant theorem on Lie algebra cohomology of nilpotent subalgebra]]
−
−
−	~~==generalization of BGG resolution==~~
−	* There exist generalizations to symmetrizable Kac-Moody algebras, cf. [34].
−	* Kempf obtained a resolution of finite-dimensional L(λ) in terms of the Grothendieck-Cousin complex in [26], which is dual to the BGG resolution.
−	* This was extended by Kumar to arbitrary Kac-Moody algebras; he thus obtained the BGG resolution here, and computed the Weyl-Kac character formula and the cohomologies of n+ (cf. [30,§9.3]).
−
−	~~==related items==~~
−	* [[Talk on BGG resolution]]
−	* [[Verma modules]]
−	* [[BGG reciprocity]]
−	* [[BGG category]]
−	* [[Kostant theorem on Lie algebra cohomology of nilpotent radical]]
−	* [[Bott-Borel-Weil Theorem]]
−	* [[Koszul complex]]
−
−	~~==books==~~
−	* [30] Shrawan Kumar, Kac-Moody Groups, their Flag Varieties and Representation Theory, Birkhauser, Progress in Math. 204, Boston, 2002
−	* James E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category O, Grad. Stud. Math., 94, Amer. Math. Soc., Providence, RI, 2008.
−
−
−	~~==expositions==~~
−	* http://rvirk.com/notes/student/catObasics.pdf
−	* BGG resolution http://www.math.columbia.edu/~woit/LieGroups-2012/vermamodules.pdf
−	* Wang, Jing Ping. “Representations of sl(2,C) in the BGG Category O and Master Symmetries.” arXiv:1408.3437 [nlin], August 14, 2014. http://arxiv.org/abs/1408.3437.
−	* http://stanford.edu/~khare/EoM-BGG-O.pdf
−
−	~~==articles==~~
−	* Pierre Julg, The Bernstein-Gelfand-Gelfand complex for rank one semi simple Lie groups as a Kasparov module, arXiv:1605.07408 [math.OA], May 24 2016, http://arxiv.org/abs/1605.07408
−	* Griffeth, Stephen, and Emily Norton. “Character Formulas and Bernstein-Gelfand-Gelfand Resolutions for Cherednik Algebra Modules.” arXiv:1511.00748 [math], November 2, 2015. http://arxiv.org/abs/1511.00748.
−	* Zelevinski, Resolvents, dual pairs, and character formulas http://www.ms.unimelb.edu.au/~ram/Resources/ResolventsDualPairsAndCharacterFormulas.html
−	* [34] A. Rocha-Caridi, Splitting Criteria for $\mathfrak{g}$-modules induced from a parabolic and the Bernstein-Gelfand-Gelfand resolution of a finite-dimensional, irreducible $\mathfrak{g}$-module, Trans. Amer. Math. Soc.262 (1980), no. 2, 335–366
−	* [26] G. Kempf, The Grothendieck-Cousin complex of an induced representation , Advances in Mathematics 29 (1978), 310–396
−	* [31] Lepowsky, J. “A Generalization of the Bernstein-Gelfand-Gelfand Resolution.” Journal of Algebra 49, no. 2 (1977): 496–511.
−	* J. Bernstein, I. Gel'fand, and S. Gel'fand, A category of g-modules, Functional Anal. Appl. 10 (1976), 87-92
−	* [5] Bernšteĭn, I. N., I. M. Gel'fand, and S. I. Gel'fand. ‘Differential Operators on the Base Affine Space and a Study of $\mathfrak{g}$-Modules’. In Lie Groups and Their Representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), 21–64. Halsted, New York, 1975. http://www.ams.org/mathscinet-getitem?mr=0578996.
−	* Bernšteĭn, I. N., I. M. Gel'fand, and S. I. Gel'fand. ‘Structure of Representations That Are Generated by Vectors of Highest Weight’. Akademija Nauk SSSR. Funkcional\cprime Nyi Analiz I Ego Priloženija 5, no. 1 (1971): 1–9.
−
−	~~[[분류:Lie theory]]~~
−	~~[[분류:abstract concepts]]~~
−	~~[[분류:migrate]]~~

← 이전 판		2020년 11월 12일 (목) 11:52 판
58번째 줄:		58번째 줄:
	[[분류:Lie theory]]		[[분류:Lie theory]]
	[[분류:abstract concepts]]		[[분류:abstract concepts]]
		+	[[분류:migrate]]

@@ 46번째 줄: / 46번째 줄: @@
 ==articles==
 * Griffeth, Stephen, and Emily Norton. “Character Formulas and Bernstein-Gelfand-Gelfand Resolutions for Cherednik Algebra Modules.” arXiv:1511.00748 [math], November 2, 2015. http://arxiv.org/abs/1511.00748.
 * Zelevinski, Resolvents, dual pairs, and character formulas http://www.ms.unimelb.edu.au/~ram/Resources/ResolventsDualPairsAndCharacterFormulas.html

@@ 10번째 줄: / 10번째 줄: @@
 Let $\lambda\in \Lambda^{+}$. There is an exact sequence of Verma modules
 $$
-\to M({w_0\cdot \lambda})\to \cdots \to \bigoplus_{w\in W, \ell(w)=k}M({w\cdot \lambda})\to \cdots M({\lambda})\to L({\lambda})\to 0
+\to M({w_0\cdot \lambda})\to \cdots \to \bigoplus_{w\in W, \ell(w)=k}M({w\cdot \lambda})\to \cdots \to M({\lambda})\to L({\lambda})\to 0
 $$
 where $\ell(w)$ is the length of the Weyl group element $w$, $w_0$ is the Weyl group element

@@ 8번째 줄: / 8번째 줄: @@
 * The BGG resolution resolves a finite-dimensional simple $\mathfrak{g}$-module $L(\lambda)$ by direct sums of Verma modules indexed by weights "of the same length" in the orbit $W\cdot \lambda$
 ;thm (Bernstein-Gelfand-Gelfand Resolution).
-Fix $\lambda\in \Lambda^{+}$. There is an exact sequence of Verma modules
+Let $\lambda\in \Lambda^{+}$. There is an exact sequence of Verma modules
 $$
 \to M({w_0\cdot \lambda})\to \cdots \to \bigoplus_{w\in W, \ell(w)=k}M({w\cdot \lambda})\to \cdots M({\lambda})\to L({\lambda})\to 0
@@ 14번째 줄: / 14번째 줄: @@
 where $\ell(w)$ is the length of the Weyl group element $w$, $w_0$ is the Weyl group element
 of maximal length. Here $\rho$ is half the sum of the positive roots.
 ==applications==

← 이전 판		2016년 4월 27일 (수) 08:57 판
57번째 줄:		57번째 줄:

	[[분류:Lie theory]]		[[분류:Lie theory]]
		+	[[분류:abstract concepts]]

@@ 27번째 줄: / 27번째 줄: @@
 ==related items==
 * [[Verma modules]]
 * [[BGG reciprocity]]
@@ 32번째 줄: / 33번째 줄: @@
 * [[Kostant theorem on Lie algebra cohomology of nilpotent radical]]
 * [[Bott-Borel-Weil Theorem]]
-* [[Talk on BGG resolution]]
+* [[Koszul complex]]
 ==books==

← 이전 판		2016년 4월 27일 (수) 08:42 판
15번째 줄:		15번째 줄:
	of maximal length. Here $\rho$ is half the sum of the positive roots.		of maximal length. Here $\rho$ is half the sum of the positive roots.

−	~~==overview==~~
−	* Weyl character formula and characters of Verma modules
−	* property of character map on short exact sequences
−	* Euler-Poincare mapping
−	* principal block : filtering through central characters
−	** is a block a $U(\mathfrak{g})$-submodule? yes
−	** how to check that it preserves the exactness : any homomorphism between modules belonging to different blocks will be zero
−	* combinatorial results
−	** consider the set of sum of k distinct roots. Which elements are linked to $0$?
−	** Bruhat ordering
−	* Bruhat ordering and strong linkage relation
−	** let $\lambda \in \Lambda^+$ (which is regular for the dot-action of $W$)
−	** $w'\cdot \lambda< w \cdot \lambda $ translates into $w < w'$ in the Bruhat ordering
−	* strong linkage relation and extension of Verma modules
−
−	~~==characters==~~
−	* let $\lambda\in \mathfrak{h}^*$
−	$$
−	~~\operatorname{ch} M({\lambda})=\frac{e^{\lambda}}{\prod_{\alpha>0}(1-e^{-\alpha})}~~
−	$$
−	* let $\lambda\in \Lambda^+$
−	~~;thm ([[Weyl-Kac character formula\|Weyl character formula]])~~
−	~~:<math>~~
−	~~\operatorname{ch}L({\lambda})=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w\cdot \lambda}}{\prod_{\alpha>0}(1-e^{-\alpha})}~~
−	~~</math>~~
−	* thus we have
−	$$
−	~~\operatorname{ch}L(\lambda)=\sum_{w \in W}(-1)^{\ell(w)}\operatorname{ch} M(w\cdot \lambda)~~
−	$$
−
−	~~;prop~~
−	~~If $0\to M' \to M \to M'' \to 0$ is a short exact sequence in $\mathcal{O}$, we have~~
−	$$
−	~~\operatorname{ch}M=\operatorname{ch}M'+\operatorname{ch}M''~~
−	$$
−
−	* if we have a long exact sequence, we still have a similar alternating sum = 0
−
−	~~;thm (Bernstein-Gelfand-Gelfand Resolution).~~
−	~~Fix $\lambda\in \Lambda^{+}$. There is an exact sequence of Verma modules~~
−	$$
−	~~0 \to M({w_0\cdot \lambda})\to \cdots \to \bigoplus_{w\in W, \ell(w)=k}M({w\cdot \lambda})\to \cdots M({\lambda})\to L({\lambda})\to 0~~
−	$$
−	~~where $\ell(w)$ is the length of the Weyl group element $w$, $w_0$ is the Weyl group element~~
−	~~of maximal length. Here $\rho$ is half the sum of the positive roots.~~
−
−	~~==example of BGG resolution==~~
−	~~===$\mathfrak{sl}_2$===~~
−
−	* <math>L({\lambda})</math> : irreducible highest weight module
−	* <math>M({\lambda})</math> : Verma modules
−	** note that the Verma modules are free modules of rank 1 over <math>\mathbb{C}[F]</math> where $F$ is the annihilation operator of $\mathfrak{sl}_2$
−	* <math>\lambda ,-2+\lambda ,\cdots, -\lambda, -\lambda-2,\cdots</math>
−	* <math>L({\lambda})=M({\lambda})/M({-\lambda-2})</math>
−	* BGG resolution
−	~~:<math>0\to M({-\lambda-2})\to M({\lambda})\to L({\lambda})\to 0</math>~~
−	* number of modules = 2 (=order of Weyl group in general)
−	* character of $L({\lambda})$ = alternating sum of characters of Verma modules
−	~~:<math>\chi_{L({\lambda})}=\chi_{M({\lambda})}-\chi_{M({-\lambda-2})}=\frac{q^{\lambda}}{1-q^{-2}}-\frac{q^{-\lambda-2}}{1-q^{-2}}</math>~~
−	* comparison with [[Weyl-Kac character formula]]
−	:<math>\chi(L({\lambda}))=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w(\lambda+\rho)}}{e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})}=\frac{q^{\lambda+1}-q^{-\lambda-1}}{q^{1}(1-q^{-2})}</math> where I used <math>\rho=1,\alpha=2</math> and <math>w(\lambda+\rho)=-\lambda-\rho</math>
−
−	~~===$\mathfrak{sl}_3$===~~
−	* http://www.math.columbia.edu/~woit/LieGroups-2012/vermamodules.pdf
−
−	~~==Euler-Poincare characteristic==~~
−	* $A$ : abelian category
−	* Euler-Poincare map $\varphi$ on an object in $A$
−	* $\varphi$ turns a short exact sequence into an alternating sum
−	* we can define Euler-Poincare characteristic $\chi_{\varphi}$ of a complex as the alternating sum of Euler-Poincare map on the homology
−	* the main result is that the Euler-Poincre characteristic can be computed from a different resolution and it is independent of the choice of it
−
−	~~==Verma modules==~~
−	~~===maps between Verma modules===~~
−
−	* 2 conditions to have non-zero homomorphisms <math>V_{\lambda}\to V_{\mu}</math> between two Verma modules
−	** <math>\lambda+\rho, \mu+\rho</math> are in the same orbit of Weyl group
−	** <math>V_{\lambda}\leq V_{\mu}</math>, i.e. <math>\lambda = \mu -\sum \alpha</math>, where the sum is over some positive roots.
−	~~====example====~~
−	* SL2
−	** <math>\lambda = \mu -2n</math>, <math>n=0,1,2,\cdots</math>
−	** <math>(\lambda+1)^2 = (\mu+1)^2</math>
−
−
−	~~===composition series of Verma modules===~~
−	~~;thm~~
−	~~The Verma module $M(\lambda)$ has a finite composition series~~
−	$$
−	~~M(\lambda)=N_0\supset N_1\supset N_2\supset \cdots N_{r}=O~~
−	$$
−	~~where each $N_i$ is a submodule of $M(\lambda)$ and $N_{i+1}$ is a maximal submodule of $N_i$. Moreover, $N_i/N_{i+1}$ is isomorphic to $L(w\cdot \lambda)$ for some $w\in W$.~~
−
−	~~===action of center on Verma modules===~~
−	* check
−
−
−	~~===maximal submodule of Verma modules===~~
−	* Maximal Submodule of $M(\lambda), \lambda \in \Lambda+$ (see 2.6)
−
−	~~==weak BGG resolution==~~
−	~~===standard filtration===~~
−	* We say that $M \in O$ has a standard filtration (also sometimes called a Verma flag) if there is a sequence of submodules
−	~~$$0 = M_0 \subset M_1 \subset M_2 \subset \cdots \subset M_n = M$$~~
−	~~for which each $M^i := M_i/M_{i−1}\, (1 \le i \le n)$ is isomorphic to a Verma module.~~
−	~~;thm (Weak BGG resolution)~~
−	~~There is an exact sequence~~
−	~~$$0 = D_m^{\lambda} \to \subset D_{m-1}^{\lambda} \to \cdots \to D_2^{\lambda} \to D_1^{\lambda} \to L(0) \to 0$$~~
−	~~where $D_{k}^{\lambda}$ has a standard filtration involving exactly once each of the Verma modules $M(w\cdot \lambda)$ with $\ell(w)=k$~~
−
−	* we prove this for $\lambda=0$ and apply the translation function to extend it
−
−	~~===strategy of the proof===~~
−	* The sequence of modules $D_k:=U(\mathfrak{g})\otimes_{U(\mathfrak{b})}\Lambda^{k}(\mathfrak{g}/\mathfrak{b})$ is a relative version of the standard resolution of the trivial module in [[Lie algebra cohomology]]
−
−	~~===standard resolution of trivial module===~~
−	* free $U(\mathfrak{g})$-modules $U(\mathfrak{g})\otimes_{\mathbb{C}}\Lambda^{k}(\mathfrak{g})$
−	* standard resolution of trivial module
−	$$\cdots \to U(\mathfrak{g})\otimes_{\mathbb{C}}\Lambda^{p}(\mathfrak{g})\to U(\mathfrak{g})\otimes_{\mathbb{C}}\Lambda^{p-1}(\mathfrak{g})\to \cdots \to U(\mathfrak{g})\otimes_{\mathbb{C}}\Lambda^{0}(\mathfrak{g})\to L(0)$$
−	* $D_k$ are free only over $U(\mathfrak{n}^{−})$
−	~~===weights of exterior powers===~~
−	* what are the weights of $\wedge^k (\mathfrak{g}/\mathfrak{b})$ as $\mathfrak{b}$-module?
−	* let $\beta$ be a sum of $k$-distinct negative root. Can we find $w\in W$ such that $w\cdot 0=\beta$?
−	* note that $\|W\cdot 0\|=\|W\|$
−	* yes, if and only if $\beta$ is a sum of elements in $w \Phi^+ \cap \Phi^-$
−
−	~~===extensions of Verma modules===~~
−	* $\mu, \lambda\in \mathfrak{h}^{*}$
−	* $\mu \uparrow \lambda$ if $\mu = \lambda$ or there is a root $\alpha$ such that $\mu=s_{\alpha}\cdot \lambda < \lambda $
−	* we say $\mu$ is strongly linked to $\lambda$ if $\mu = \lambda$ or there exist root $\alpha_1,\cdots, \alpha_r\in \Phi^+$ such that $\mu=(s_{\alpha_1}\cdots s_{\alpha_r})\cdot \lambda \uparrow (s_{\alpha_2}\cdots s_{\alpha_r})\cdot \lambda \uparrow ( s_{\alpha_r})\cdot \lambda \uparrow \cdots \uparrow \lambda $
−	~~;thm~~
−	~~Let $\lambda\in \mathfrak{h}^{*}$.~~
−
−	~~(a) If $\operatorname{Ext}_{\mathcal{O}}(M(\mu),M(\lambda))\neq 0$ for $\mu\in \mathfrak{h}^{*}$, then $\mu \uparrow \lambda$ but $\mu \neq \lambda$~~
−
−	(b) Let $\lambda\in \Lambda^{+}$ and $w,w'\in W$. If $\operatorname{Ext}_{\mathcal{O}}(M(w'\cdot\lambda),M(w\cdot\lambda))\neq 0$, then $w<w'$ in the [[Bruhat ordering]]. In particular, $\ell(w)<\ell(w')$.

	==applications==		==applications==

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
 * Weyl character formula. For $\lambda\in \Lambda^{+}$,
 $$