S. Sundaram, V. Welker, Group actions on arrangements of linear subspaces and applications to configuration spaces, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1389--1420.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....loopfree categories. In Section 4 we draw connections to determining the multiplicity of the trivial character in the induced representations of G on the homology groups of the nerve of the category, derive a formula for the Mobius function of P G and, based on formulae of Sundaram and Welker, [18], give a quotient analog of GoreskyMacPherson formulae. As another example where these methods proved to be essential we would like to mention the computation of the homology groups of the deleted symmetric join of an infinite simplex, see [1] Date: December 9, 2001 Mathematical Subject ....

S. Sundaram, V. Welker, Group actions on arrangements of linear subspaces and applications to configuration spaces, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1389--1420.

....resp. monic complex, polynomials of degree n with specified root multiplicities. These spaces naturally appear in singularity theory, AGV85] Homological invariants of several of these strata were in particular computed by Arnol d, Shapiro, Sundaram, Welker, Vassiliev, and the author, see [Ar70a, Ko99a, Ko00b, ShW98, SuW97, Vas98]. 1.1.2 The idea of the resonance category and resonance functors The purpose of the research presented in this chapter is to take a different, more abstract look at this set of problems. More specifically, the idea is to introduce a new canonical combinatorial object, independent of the ....

....we choose to use a class of topological spaces which come in particular from singularity theory, and whose topological properties have been studied: spaces of polynomials (real or complex) with prescribed root multiplicities. In particular, in case of strata (k ) which were studied in [Ar70a, Ko99a, SuW97] for the complex case, and in [Ko00b, ShW98] for the real case, we demonstrate how the inherent combinatorial structure of the resonance category makes this particular resonance especially reducible. Here is the brief summary of the contents of this chapter. Section 1.2. We introduce the ....

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S. Sundaram, V. Welker, Group actions on arrangements of linear subspaces and applications to configuration spaces, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1389--1420.

....from the formulas of Section 4. They can also be explained by considering the variant of doubly indexed Whitney homology on Pi n . We also touch upon connections with subspace arrangements. We use the formulas of Section 4 and an equivariant Goresky MacPherson formula of Sundaram and Welker [SWe1] to derive a formula for the representation of the symmetric group on the cohomology of the complement of a complexified subspace arrangement whose intersection lattice is Pi n . 1. Preliminaries. Let P be a finite bounded poset of length = P ) 0, with minimum element 0 and maximum ....

....lattice first arose in the work of Bjorner Lov asz and Yao [BLY] in connection with a computational complexity problem. Its homology was further studied in papers by Bjorner and Lov asz [BL] Bjorner and Welker [BWe] Bjorner and Wachs [BW2] Sundaram and Wachs [SW] and Sundaram and Welker [SWe1]. Example 3.16 suggests that perhaps additivity can be dropped from the hypothesis of Theorem 3.14. This, however, is not the case as Example 3.17 below shows. Example 3.18 shows that reductivity cannot be dropped either. Hence neither additivity nor reductivity alone is sufficient for ....

[Article contains additional citation context not shown here]

S. Sundaram and V. Welker, Group actions on arrangements of linear subspaces and applications to configuration spaces, preprint 1994, Trans. AMS (to appear).

....Betti numbers of spaces Sigma . In [2] V.I. Arnold has computed fi ( Sigma ; Q) for = k m ; 1 n Gammakm ) Theorem 1.2. 2] Let = k m ; 1 n Gammakm ) for some natural numbers k 2, m, and n km. Then fi i ( Sigma ; Q) 1; for i = 2l( 0; otherwise. 1. 1) In [18] Sundaram and Welker conjectured that Conjecture 1.3. For any number partition , fi i ( Sigma ; Q) 0 unless i = 2l( In this paper we shall give a new, combinatorial proof of the Theorem 1.2 and disprove Conjecture 1.3. Date: September 24, 1999 Research at IAS was supported by von ....

....is along the lines of [15] which has been extended to cover the result of Arnold and clarified by using the combinatorial cell description, from Section 4, of X ; We would like to mention that for the special cases = k; 1 n Gammak ) and = k m ) the Theorem 1. 2 was also reproved in [18] by using the Theorem 3.2 (as Theorem 6.6 shows, the case = k m ) is especially simple) However, the Theorem 5.2 is the first combinatorial (modulo Theorem 3.2) proof of the result of Arnold in the general case. In Section 6 we disprove the conjecture of Sundaram and Welker. Besides giving a ....

[Article contains additional citation context not shown here]

S. Sundaram, V. Welker, Group actions on arrangements of linear subspaces and applications to configuration spaces, Trans. Amer. Math. Soc. 349, no. 4, (1997), pp. 1389--1420.

....[BK1, BK2] Now we are ready for the nonpure case. Bjorner and Wachs generalization of shellability to nonpure simplicial complexes, made by simply dropping the assumption of purity [BW2, BW3] generated a great deal of interest, and sparked the generalization of several other related concepts [SWa, SWe, BS, DR]. In particular, Stanley introduced sequential Cohen Macaulayness [St6, Section III.2] a nonpure generalization of Cohen Macaulayness, and designed the (algebraic) definition so that a nonpure shellable complex is sequentially Cohen Macaulay, much as a shellable complex is Cohen Macaulay. ....

S. Sundaram and V. Welker, "Group actions on arrangements of linear subspaces and applications to configuration spaces," Trans. Amer. Math. Soc.,toappear.

....from the formulas of Section 4. They can also be explained by considering the variant of doubly indexed Whitney homology on Pi T n . We also touch upon connections with subspace arrangements. We use the formulas of Section 4 and an equivariant Goresky MacPherson formula of Sundaram and Welker [SWe1] to derive a formula for the representation of the symmetric group on the cohomology of the complement of a complexified subspace arrangement whose intersection lattice is Pi T n . 4 MICHELLE L. WACHS 1. Preliminaries. Let P be a finite bounded poset of length = P ) 0, with minimum ....

....lattice first arose in the work of Bjorner Lov asz and Yao [BLY] in connection with a computational complexity problem. Its homology was further studied in papers by Bjorner and Lov asz [BL] Bjorner and Welker [BWe] Bjorner and Wachs [BW2] Sundaram and Wachs [SW] and Sundaram and Welker [SWe1]. WHITNEY HOMOLOGY OF SEMIPURE SHELLABLE POSETS 17 Example 3.16 suggests that perhaps additivity can be dropped from the hypothesis of Theorem 3.14. This, however, is not the case as Example 3.17 below shows. Example 3.18 shows that reductivity cannot be dropped either. Hence neither additivity ....

[Article contains additional citation context not shown here]

S. Sundaram and V. Welker, Group actions on arrangements of linear subspaces and applications to configuration spaces, preprint 1994, Trans. AMS (to appear).

....[BK1, BK2] Now we are ready for the nonpure case. Bjorner and Wachs generalization of shellability to nonpure simplicial complexes, made by simply dropping the assumption of purity [BW2, BW3] generated a great deal of interest, and sparked the generalization of several other related concepts [SWa, SWe, BS, DR]. In particular, Stanley introduced sequential Cohen Macaulayness [St6, Section III.2] a nonpure generalization of Cohen Macaulayness, and designed the (algebraic) definition so that a nonpure shellable complex is sequentially Cohen Macaulay, much as a shellable complex is Cohen Macaulay. ....

S. Sundaram and V. Welker, "Group actions on arrangements of linear subspaces and applications to configuration spaces," Trans. Amer. Math. Soc., to appear.

....Frobenius characteristic of the Sn action on the Whitney homology ( Bac] Bj2] that is, the homology of lower intervals of Pi (k;1 n Gammak ) The result of this computation (Theorem 3. 7) is more intricate than the expression for the homology of Pi (k;1 n Gammak ) however, as shown in [SW], it is an essential ingredient in determining the representation of Sn on the cohomology of the complement of the k equal arrangement. In particular in [SW] this computation is applied to the study of the related orbit space. The characters which arise in this paper are closely related to the ....

....result of this computation (Theorem 3.7) is more intricate than the expression for the homology of Pi (k;1 n Gammak ) however, as shown in [SW] it is an essential ingredient in determining the representation of Sn on the cohomology of the complement of the k equal arrangement. In particular in [SW] this computation is applied to the study of the related orbit space. The characters which arise in this paper are closely related to the character of Sn on the multilinear part of the free Lie algebra. For k = 2 the basis for cohomology provided by the shelling gives an Sn equivariant ....

[Article contains additional citation context not shown here]

Sundaram, S. and Welker, V.: Groups actions on arrangements of linear subspaces and applications to configuration spaces, Trans. Amer. Math. Soc., to appear.

....which allows one to replace limits by homotopy limits, and the Homotopy Lemma [37] 6, XII.4.2] 39] which compares the homotopy types of diagrams over the same partially ordered set. These tools have found striking applications, for example, in the study of subspace arrangements [44] [36]. Our objective here is to provide a tool kit for manipulation of limits and homotopy limits: basic lemmas that allow one to compare the homotopy types of diagrams defined over different partial orders. Not all of these results are new: in particular, Dwyer Kan [13, Sect. 9] provide a list of ....

S. Sundaram and V. Welker. Group actions on arrangements of linear subspaces and applications to configuration spaces. Preprint, 1994.

....which sometimes allows one to replace colimits by homotopy colimits, and the Homotopy Lemma [40] 6, XII.4.2] 42] which compares the homotopy types of diagrams over the same small category. These tools have found striking applications, for example, in the study of subspace arrangements [47] [39] [34] The choice of contents for our toolkit for the manipulation of homotopy colimits is partially motivated by the usefulness of corresponding lemmas in the special case of order complexes (the discrete case, when the spaces of the diagram are points) In this case, there is a solid amount of ....

S. Sundaram and V. Welker. Group actions on arrangements of linear subspaces and applications to configuration spaces. Transactions Amer. Math. Soc., 349:1389--1420, 1997.

....Frobenius characteristic of the Sn action on the Whitney homology ( Bac] Bj2] that is, the homology of lower intervals of Pi (k;1 n Gammak ) The result of this computation (Theorem 3. 7) is more intricate than the expression for the homology of Pi (k;1 n Gammak ) however, as shown in [SW], it is an essential ingredient in determining the representation of Sn on the cohomology of the complement of the k equal arrangement. In particular in [SW] this computation is applied to the study of the related orbit space. The characters which arise in this paper are closely related to the ....

....result of this computation (Theorem 3.7) is more intricate than the expression for the homology of Pi (k;1 n Gammak ) however, as shown in [SW] it is an essential ingredient in determining the representation of Sn on the cohomology of the complement of the k equal arrangement. In particular in [SW] this computation is applied to the study of the related orbit space. The characters which arise in this paper are closely related to the character of Sn on the multilinear part of the free Lie algebra. For k = 2 the basis for cohomology provided by the shelling gives an Sn equivariant ....

[Article contains additional citation context not shown here]

Sundaram, S. and Welker, V.: Groups actions on arrangements of linear subspaces and applications to configuration spaces, Trans. Amer. Math. Soc., to appear.

No context found.

S. Sundaram and V. Welker, Group actions on arrangements of linear subspaces and applications to configuration spaces, preprint 1994, Trans. Amer. Math. Soc. (to appear).

No context found.

S. Sundaram and V. Welker, Group actions on arrangements of linear subspaces and applications to configuration spaces, preprint 1994, Trans. Amer. Math. Soc. (to appear).

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