Mary Schaps

DEFORMATIONS OF COHEN-MACAULAY SCHEMES OF CODIMENSION 2 AND NON-SINGULAR DEFORMATIONS OF SPACE CURVES.

The following work deals with the deformations of embedded affine schemes of codimension 2, which locally have a resolution of length 2. The cases of immediate interest are curves in 3-space and 0-dimensional schemes in the plane. It is first shown that such a scheme X has a global resolution of length 2. Therefore, by a theorem of Burch, the functions defining the ideal of X can be obtained as the maximal minors of a matrix whose columns generate all the relations among these functions. All flat deformations of X can be obtained simply by deforming this matrix, and this permits the construction of the versal deformation space of X. Finally, for X of dimension 3 or less one can construct non-singular deformations of X by taking a parameter space sufficiently large to permit one to change the constant and linear terms of each entry in the matrix. For X of dimension 4, an example is given in XXX [11] in which the scheme not only has no non-singular deformations, but in fact has no non-isomorphic deformations at all. A brief review of the previous literature will help place these results in perspective. It has long been known, by Bertinis theorem, that the generic deformation of a scheme of codimension 1 is non-singular. As a consequence of the work of Fogarty [5], it was also known that every point, or rather, 0-dimensional scheme in the plane has non-singular deformations; Briancon and Galligo [3] give an explicit construction for such a deformation, splitting the scheme into distinct simple points. This led mathematicians interested in algebraic curves to ask if the generic deformation of a space curve is also non-singular, the question which is settled in this paper. Further direct extension of these results in the case of curves is impossible, since the work of Iarrobina [4] permits the construction of a non-reduced curve in affine 4-space which has no non-singular deformations; however, for reduced curves the question is still open as of this writing.

Crossover morita equivalences for blocks of the covering groups of the symmetric and alternating groups

In [S], Joanna Scopes discovered a method for generating Morita equivalences between blocks of symmetric groups and thus for showing that Donavan’s conjecture, that there are only a finite number of Morita equivalence classes of blocks with a given defect group, holds for the blocks of the symmetric groups. This method has led in various different directions. It was generalized by Puig [P1] to demonstrate not only Morita equivalences but also the more restrictive Puig equivalences, thus establishing Puig’s conjecture, that there are only a finite number of Puig equivalence classes for a given defect group, for blocks of the symmetric group. A variant was adapted by the first author to prove Donovan’s conjecture for blocks of the Schur covers of the symmetric and alternating groups, [K]. A related technique was used in [Jo] for blocks of the general linear group, and an adaptation of the method was developed in [HK1], [HK2] to find Morita equivalences between blocks in various other algebraic groups. The method also lead Rickard to a way of demonstrating derived quivalences between blocks of symmetric groups, and this method was then taken up by Chuang and Rouquier [ChR] to show that for a given weight there is only one derived equivalence class of symmetric blocks which, along with [ChK], settled the Broue conjecture for symmetric blocks. In this paper we intend to return to [K] and show that, in fact, the results therein reflected only half of the picture. The results in [K] demonstrated the existence of Morita equivalences between blocks of the covering groups S̃n of Sn or between blocks of the covering groups Ãn of An. We will now reconsider the situation and show that we can equally well get “crossovers” between blocks of Ãn and S̃n. More specifically, the various characters are associated with strict partitions of n and the Morita eqivalences are obtained by an involution which is a variant of the Scopes involution used in Scopes’ original work. The cases treated in [K] were those in which the involution is parity-preserving, and in this paper we will be interested in cases where it is parity-reversing.

Maximal Strings in the Crystal Graph of Spin Representations of the Symmetric and Alternating Groups

We define a block-reduced version of the crystal graph of spin representations of the symmetric and alternating groups, and separate it into layers, each obtained by translating the previous layer and, possibly, adding new defect zero blocks. We demonstrate that each layer has weight-preserving central symmetry, and study the sequence of weights occurring in the maximal strings. The Broué conjecture, that a block with abelian defect group is derived equivalent to its Brauer correspondent, has been proven for blocks of cyclic defect group and verified for many other blocks. This article is part of a study of the spin block case.

The Donald–Flanigan Problem for Finite Reflection Groups

AbstractThe Donald–Flanigan problem for a finite group H and coefficient ring k asks for a deformation of the group algebra kH to a separable algebra. It is solved here for dihedral groups and Weyl groups of types Bn and Dn (whose rational group algebras are computed), leaving but six finite reflection groups with solutions unknown. We determine the structure of a wreath product of a group with a sum of central separable algebras and show that if there is a solution for H over k which is a sum of central separable algebras and if Sn is the symmetric group then (i) the problem is solvable also for the wreath product H

The modular version of Maschke's theorem for normal abelian p-Sylows

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FINITE POSETS AND THEIR REPRESENTATION ALGEBRAS

Sn = H × ··· × H (n times) ⋊ Sn and (ii) given a morphism from a finite Abelian or dihedral group G to Sn it is solvable also for H

Deformations of Algebras and Hochschild Cohomology

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