We provide an algorithmic description of a family of graded decomposition numbers for diagrammati... more We provide an algorithmic description of a family of graded decomposition numbers for diagrammatic Cherednik algebras in terms of affine Kazhdan-Lusztig polynomials.
We construct a family of graded isomorphisms between certain subquotients of diagrammatic Cheredn... more We construct a family of graded isomorphisms between certain subquotients of diagrammatic Cherednik algebras as the quantum characteristic, multicharge, level, degree, and weighting are allowed to vary; this provides new structural information even in the case of the classical q-Schur algebra. This also allows us to prove some of the first results concerning the (graded) decomposition numbers of these algebras over fields of arbitrary characteristic.
In this paper we describe the blocks of the partition algebra over a field of positive characteri... more In this paper we describe the blocks of the partition algebra over a field of positive characteristic.
We introduce Brauer algebras associated to complex reflection groups of type G(m, p, n), and stud... more We introduce Brauer algebras associated to complex reflection groups of type G(m, p, n), and study their representation theory via Clifford theory. In particular, we determine the decomposition numbers of these algebras in characteristic zero.
We propose a new approach to study the Kronecker coefficients by using the Schur-Weyl duality bet... more We propose a new approach to study the Kronecker coefficients by using the Schur-Weyl duality between the symmetric group and the partition algebra. We explain the limiting behavior and associated bounds in the context of the partition algebra. Our analysis leads to a uniform description of the Kronecker coefficients when one of the indexing partitions is a hook or a two-part partition. France
We study the representation theory of the cyclotomic Brauer algebra via truncation to idempotent ... more We study the representation theory of the cyclotomic Brauer algebra via truncation to idempotent subalgebras which are isomorphic to a product of walled and classical Brauer algebras. In particular, we determine the block structure and decomposition numbers in characteristic zero.
We study the structure of the indecomposable direct summands of tensor products of two restricted... more We study the structure of the indecomposable direct summands of tensor products of two restricted rational simple modules for the algebraic group SL3(K), where K is an algebraically closed field of characteristic p ≥ 5. We also give a characteristicfree algorithm for the decomposition of such a tensor product into indecomposable direct summands. The p < 5 case was studied in the authors' earlier paper . We find that for characteristics p ≥ 5 all the indecomposable summands are rigid, in contrast to the characteristic 3 case.
We extend the the combinatorics of tableaux to the study of Brauer, walled Brauer, and partition ... more We extend the the combinatorics of tableaux to the study of Brauer, walled Brauer, and partition algebras. In particular, we provide uniform constructions of Murphy bases and 'Specht' filtrations of permutation modules. This allows us to give a uniform construction of semistandard bases of their quasi-hereditary covers.
In a recent paper Cohen, Liu and Yu introduce the Type C Brauer algebra. We show that this algebr... more In a recent paper Cohen, Liu and Yu introduce the Type C Brauer algebra. We show that this algebra is an iterated inflation of hyperoctahedral groups, and that it is cellularly stratified. This gives an indexing set of the standard modules, results on decomposition numbers, and the conditions under which the algebra is quasi-hereditary.
We show that an adaptation of Landrock’s Lemma for symmetric algebras also holds for cellular alg... more We show that an adaptation of Landrock’s Lemma for symmetric algebras also holds for cellular algebras and BGG algebras. This is a result relating the radical layers of any two projective modules. As a corollary we deduce that BGG reciprocity respects Loewy structure.
We give an algorithm for working out the indecomposable direct summands in a Krull–Schmidt decomp... more We give an algorithm for working out the indecomposable direct summands in a Krull–Schmidt decomposition of a tensor product of two simple modules for SL3 in characteristics 2 and 3. It is shown that there is a finite family of modules such that every such indecomposable summand is expressible as a twisted tensor product of members of that family.
We provide an algorithmic description of a family of graded decomposition numbers for diagrammati... more We provide an algorithmic description of a family of graded decomposition numbers for diagrammatic Cherednik algebras in terms of affine Kazhdan-Lusztig polynomials.
We construct a family of graded isomorphisms between certain subquotients of diagrammatic Cheredn... more We construct a family of graded isomorphisms between certain subquotients of diagrammatic Cherednik algebras as the quantum characteristic, multicharge, level, degree, and weighting are allowed to vary; this provides new structural information even in the case of the classical q-Schur algebra. This also allows us to prove some of the first results concerning the (graded) decomposition numbers of these algebras over fields of arbitrary characteristic.
In this paper we describe the blocks of the partition algebra over a field of positive characteri... more In this paper we describe the blocks of the partition algebra over a field of positive characteristic.
We introduce Brauer algebras associated to complex reflection groups of type G(m, p, n), and stud... more We introduce Brauer algebras associated to complex reflection groups of type G(m, p, n), and study their representation theory via Clifford theory. In particular, we determine the decomposition numbers of these algebras in characteristic zero.
We propose a new approach to study the Kronecker coefficients by using the Schur-Weyl duality bet... more We propose a new approach to study the Kronecker coefficients by using the Schur-Weyl duality between the symmetric group and the partition algebra. We explain the limiting behavior and associated bounds in the context of the partition algebra. Our analysis leads to a uniform description of the Kronecker coefficients when one of the indexing partitions is a hook or a two-part partition. France
We study the representation theory of the cyclotomic Brauer algebra via truncation to idempotent ... more We study the representation theory of the cyclotomic Brauer algebra via truncation to idempotent subalgebras which are isomorphic to a product of walled and classical Brauer algebras. In particular, we determine the block structure and decomposition numbers in characteristic zero.
We study the structure of the indecomposable direct summands of tensor products of two restricted... more We study the structure of the indecomposable direct summands of tensor products of two restricted rational simple modules for the algebraic group SL3(K), where K is an algebraically closed field of characteristic p ≥ 5. We also give a characteristicfree algorithm for the decomposition of such a tensor product into indecomposable direct summands. The p < 5 case was studied in the authors' earlier paper . We find that for characteristics p ≥ 5 all the indecomposable summands are rigid, in contrast to the characteristic 3 case.
We extend the the combinatorics of tableaux to the study of Brauer, walled Brauer, and partition ... more We extend the the combinatorics of tableaux to the study of Brauer, walled Brauer, and partition algebras. In particular, we provide uniform constructions of Murphy bases and 'Specht' filtrations of permutation modules. This allows us to give a uniform construction of semistandard bases of their quasi-hereditary covers.
In a recent paper Cohen, Liu and Yu introduce the Type C Brauer algebra. We show that this algebr... more In a recent paper Cohen, Liu and Yu introduce the Type C Brauer algebra. We show that this algebra is an iterated inflation of hyperoctahedral groups, and that it is cellularly stratified. This gives an indexing set of the standard modules, results on decomposition numbers, and the conditions under which the algebra is quasi-hereditary.
We show that an adaptation of Landrock’s Lemma for symmetric algebras also holds for cellular alg... more We show that an adaptation of Landrock’s Lemma for symmetric algebras also holds for cellular algebras and BGG algebras. This is a result relating the radical layers of any two projective modules. As a corollary we deduce that BGG reciprocity respects Loewy structure.
We give an algorithm for working out the indecomposable direct summands in a Krull–Schmidt decomp... more We give an algorithm for working out the indecomposable direct summands in a Krull–Schmidt decomposition of a tensor product of two simple modules for SL3 in characteristics 2 and 3. It is shown that there is a finite family of modules such that every such indecomposable summand is expressible as a twisted tensor product of members of that family.
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