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For the Kashiwara crystal of a highest weight representation of an affine Lie algebra of type A and rank e, with highest weight Λ, there is a labeling by multipartitions and by piecewise-linear paths in the real weight space called Littelmann paths. Both labelings are constructed recursively, but the crystals are isomorphic, so there is a bijection between the labels. We choose a multicharge $(k_{1},\dots ,k_{r})$ , with 0 ≤ k1 ≤ k2.... ≤ kr ≤ e − 1. We put ki in the node at the upper left corner of partition i and let the residues from $\mathbb {Z}/ e \mathbb {Z}$ increase across rows and decrease down columns. For e = 2, we call a multipartition residue-homogeneous if all nonzero rows end in nodes of the same residue and if partitions with the same corner residue have first rows of the same parity. The multipartition is called strongly residue-homogeneous if each partition ends in a right triangle of whose side has length one less than the first row of the next partition. We give explicit examples, including many of the important case of symmetric groups. We show that such a multipartition corresponds to a Littelmann path which is unidirectional in the sense that the projection of the the main part of the path to the coordinates of the fundamental weights consists of long paths all lying in either the second or fourth quadrant, separated by short paths between fixed integers encoding the number of rows and addable nodes in the multipartition. The path corresponding to a strongly residue-homogeneous multipartition can be constructed non-recursively using only integers describing the multipartition.

For symmetric Kashiwara crystals of type $A$ and rank $e=2$, and for the canonical basis elements that we call external, corresponding to weights on the outer skin of the Kashiwara crystal, we construct the canonical basis elements in a non-recursive manner. In particular, for a symmetric crystal with $\Lambda=a \Lambda_0+a \Lambda_1$, we give formulae for the canonical basis elements for all the $e$-regular multipartitions with defects either $k(a-k)$ or $k(a-k)+2a$, for $0 \leq k \leq a$., Comment: 21 pages, 1 figure; typos corrected

In this paper, we investigate the basis graph of the monoid algebra of a submonoid of the monoid of mappings from N = {1,…,n} to itself, defined by a nested sequence of compositions of N. Each such monoid is a left regular band (LRB), that is, a semigroup S satisfying x2 = x and xyx = xy for all x,y∈S. This class is sufficiently rich that every path algebra of an acyclic quiver can be embedded in such a monoid algebra. The multiplication in the monoid algebra has a particularly simple quasi-multiplicative form, allowing definition over the integers. Combining this with a formula for Ext-groups for LRBs due to Margolis et al. [6], we get a simple criterion for the nested composition algebras to be hereditary.

We complete the demonstration of source algebra equivalences between spin blocks of families of covering groups $\{\widetilde {S}_n\}$ and $\{\widetilde {A}_n\}$ of symmetric and alternating groups, for pairs of blocks at the ends of maximal strings. These equivalences remain within the family of groups if cores of the two blocks have the same parity and cross over from one family to the other if the cores are of opposite parity. This demonstrates Kessar and Schaps' crossover conjecture for the easier case of extremal points of maximal strings. We use this result to give an improved bound for the highest degree necessary in order to get representatives of all Morita equivalence classes of spin blocks for a given weight.

Let M be a monoid, and let L be a commutative idempotent submonoid. We show that we can find a complete set of orthogonal idempotents L^0 of the monoid algebra A of M such that there is a basis of A adapted to this set of idempotents which is in one-to-one correspondence with elements of the monoid. The basis graph describing the Peirce decomposition with respect to L^0 gives a coarse structure of the algebra, of which any complete set of primitive idempotents gives a refinement, and we give some criterion for this coarse structure to actually be a fine structure, which means that the nonzero elements of the monoid are in one-to-one correspondence with the vertices and arrows of the basis graph with respect to a set of primitive idempotents, with this basis graph being a canonical object.

Let Λ be a dominant integral weight of level k for the affine Lie algebra g and let α be a non-negative integral combination of simple roots. We address the question of whether the weight η = Λ − α lies in the set P(Λ) of weights in the irreducible highest-weight module with highest weight Λ. We give a non-recursive criterion in terms of the coefficients of α modulo an integral lattice kM, where M is the lattice parameterizing the abelian normal subgroup T of the Weyl group. The criterion requires the preliminary computation of a set no larger than the fundamental region for kM, and we show how this set can be efficiently calculated.

We clarify the explicit structure of the Hurwitz quaternion order, which is of fundamental importance in Riemann surface theory and systolic geometry., Comment: 14 pages, to appear in Geometriae Dedicata

A finite poset (P ≼ S) determines a finite dimensional algebra TP over the field 𝔽 of two elements, with an upper triangular representation. We determine the structure of the radical of the representation algebra A of the monoid (TP,·) over a field of characteristic different from 2. We also consider degenerations of A over a complete discrete valuation ring with residue field of characteristic 2.

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 Broue 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.

We verify Broue's conjecture for the faithful 3-blocks of defect 2 of the non-split central extension of the sporadic simple Mathieu group M 22 by a cyclic group of order 4. The proof is based on a strategy due to Okuyama and Rickard, where a stable equivalence is lifted to a derived equivalence. The stable equivalence in turn is provided by exploiting a result due to Puig. To handle this particular example, next to theoretical investigations we apply a whole bunch of computational tools.

The problem of pattern matching with rotation is that of finding all occurrences of a two-dimensional pattern in a text, in all possible rotations. We prove an upper and lower bound on the number of such different possible rotated patterns. Subsequently, given an m×m array (pattern) and an n×n array (text) over some finite alphabet Σ, we present a new method yielding an O(n2m3) time algorithm for this problem.

We show that any weak crossed product which is a specialization of a crossed product satisfies an extendability condition on the cocycle and a monomial lifting condition on its idempotent cocycle. We interpret the idempotent condition in terms of the partial ordering induced on the cosets of the associated group. The results are applied to weakly Azumaya algebras.

Based on tilting theory, we demonstrate the existence of homogeneous deformations for the Brauer tree algebras, which are derived naturally from a tilting complex.

We give dual one-sided tilting complexes producing inverse equivalences of the derived category of a Brauer star algebra and a Brauer tree algebra of the same type, folded according to an additional combinatorial structure on the Brauer tree. We relate this to the two-sided two-term tilting complex of Rouquier in the case of a group block, showing that it induces the “completely folded” case for each one-sided complex.

In this paper we establish a one-to-one correspondence between Brauer trees with an additional structure that we call “pointing” and equivalence classes of tilting complexes of the corresponding Brauer star algebra which are of a special “combinatorial” nature. Under the correspondence, the identification of the indecomposable summands of the tilting complex with edges of the Brauer tree is completely natural. The procedure gives an algorithm for producing large numbers of distinct tilting complexes for the same Brauer tree.

The 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\(\wr\)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\(\wr\)G. The theorems suggested by the Donald–Flanigan conjecture and subsequently proven follow, we also show, from a geometric conjecture which although weaker for groups applies to a broader class of algebras than group algebras.

Let K be a sufficiently large field of characteristic p. We determine the quiver of a semidirect product H = P ⋊ [Hbar] of a finite p-group [Hbar] as the join of the quiver of [Hbar] and the McKay graph D([Hbar], [Rbar]') of the conjugation representation [Rbar]' of [Hbar] on . More generally, whenever , we show that the quiver QKH of H is a subgraph of the above join, and give a necessary and sufficient condition on the radicals for the quiver of H to exactly equal the join . Finally, we identify the “transgressing” arrows of the McKay graph, those which do not appear in the quiver of H, with basis elements in the kernel of the transgression map.

We prove that the group algebra of a finite group with a cyclic p -Sylow subgroup over an algebraically closed field is a specialization of a parameter-dependent multiplication structure which gives a semisimple algebra for general values of the parameter. We actually prove the existence of such a specialization for any block of cyclic defect group.

We prove that the split integral group ring of a finite p-solvable group of finite representation type has a structure analogous to that of the p-modular semisimple deformation. The split integral deformation can be put in the same form as the p-modular deformation by an appropriate substitution for the parameter T. As an application we derive a simple formula for the matrix units in the semisimple group algebra over a nonmodular prime.

We give an algorithmic procedure for constructing the quivers and homogeneous relations of the Brauer correspondents of blocks of abelian noncyclic defect group in the almost simple groups in the ATLAS. This is one step in a program to compute the structure of the indecomposable projectives for these blocks. As an illustration, we determine an explicit tilting complex for the non-principal 3-block of the central extension of PSL(3,4) by C2.

The Broue 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, mostly with defect group C 3 × C 3 or C 5 × C 5 . In this paper, we exhibit explicit tilting complexes from the Brauer correspondent to the global block B for a number of Morita equivalence classes of blocks of defect group C 3 × C 3. We also describe a database with data sheets for over a thousand blocks of abelian defect group in the ATLAS group and their subgroups. Introduction Let G be a finite group and let k be a field of characteristic p , where p divides | G |. Let kG =⊕ B i be a decomposition of the group algebra into blocks, and let D i be the defect group of the block B i , of order. By Brauer's Main Theorems (see [1] for an accessible exposition) there is a one-to-one correspondence between blocks of kG with defect group D i and blocks of kNG(D i ) with defect group D i . Let b i be the block corresponding to B i , called its Brauer correspondent. Broue [5] has conjectured that if D i is abelian and B i is a principal block, then B i and b i are derived equivalent, i.e., the bounded derived categories D b (B i ) and D b (b i ) are equivalent. In fact, it is generally believed by researchers in the field that the hypothesis that B i be principal is unnecessary.

The concept of a generalized complete intersection (GCI) of affine schemes is introduced. The proofs of the following theorems are then sketched. Theorem 1. Suppose X X is a G C I GCI of affine Cohen-Macaulay schemes of codimension ⩽ 2 \leqslant 2 . Then if the versal deformation space of X X exists, it is smooth. Theorem 2. Suppose X X is a G C I GCI of determinantal schemes. Then if dim ⁡ X \dim X is sufficiently low, X X is smoothable. In particular, if X X is as in Theorem 1 and dim ⁡ X > 4 \dim X > 4 then X X is smoothable.

The use of computers to investigate groups has mainly been restricted to finite groups. In this work, a method is given for finding all subgroups of finite index in a given group, which works equally well for finite and for infinite groups. The basic object of study is the finite set of cosets. §2 reviews briefly the representation of a subgroup by permutations of its cosets, introduces the concept of normal coset numbering, due independently to M. Schaps and C. Sims, and describes a version of the Todd-Coxeter algorithm. §3 contains a version due to A. Dietze of a process which was communicated to J. Neubuser by C. Sims, as well as a proof that the process solves the problem stated in the title. A second such process, developed independently by M. Schaps, is described in §4. §5 gives a method for classifying the subgroups by conjugacy, and §6, a suggestion for generalization of the methods to permutation and matrix groups.

In this paper we count the number of distinct toroidal morphisms to 3-space satisfying Danilov's condition (0), obtaining the simple formulaMn=3·4n−1 forn blowings up. We also give average and extremal values for the number of alternative factorization sequences for such a morphism.

Let B B be a finite dimensional algebra over an algebraically closed field K K . If we represent primitive idempotents by points and basis vectors in e i B e j {e_i}B{e_j} by "arrows" from e j {e_j} to e i {e_i} , then any specialization of the algebra acts on this directed graph by coalescing points. This implies that the number of irreducible components in the scheme parametrizing n n -dimensional algebras is no less than the number of loopless directed graphs with a total of n n vertices and arrows. We also show that the condition of having a distributive ideal lattice is open.

If G is a finite group with abelian normal p -Sylow subgroup P , and F is a sufficiently large field of characteristic p , then the group algebra FG can be deformed as an F -algebra to a separable algebra with the same matrix components as in the characteristic zero case.

The paper introduces formally the concept of local factorizability used in earlier factorizability work, identifies the basic form of obstructions to local factorizability of birational morphisms, and outlines a combinatorial technique for analyzing such obstructions. As an application and illustration, two open cases in the classification of birational morphisms with small canonical divisors are settled.

Since Hochschild cohomology is preserved under tilting, two algebras A and B related by tilting have infinitesimal deformation spaces of the same dimension. We show how, for each deformation of A, we can construct a deformation of the tilting module ATB which will provide directly a deformation of B. We also show that the calculation of the vector space of infinitesimal deformations of a module is a feasible microcomputer calculation, and prove that this vector space is the tangent space to a versal deformation space.

We show that if a finite dimensional algebra A has both a deformation A and a tilting B = EndA(M), then we can deform the tilting module M to \( \tilde M \) in order to obtain a deformation \( \tilde B \) of B such that \( \tilde B = End_{\tilde A} (\tilde M) \) is a tilting of A.

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 Bertini's 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.