A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
The point of view of these notes on the topic is to bring out the flavour that Representation Theory is an extension of the first course on Group Theory. We …
The point of view of these notes on the topic is to bring out the flavour that Representation Theory is an extension of the first course on Group Theory. We also emphasize the importance of the base field. These notes cover completely the theory over complex numbers which is Character Theory. A large number of worked-out examples are the main feature of these notes. The prerequisite for this note is basic group theory and linear algebra.
The point of view of these notes on the topic is to bring out the flavor that Representation Theory is an extension of the first course on Group Theory. We …
The point of view of these notes on the topic is to bring out the flavor that Representation Theory is an extension of the first course on Group Theory. We also emphasize the importance of base field. These notes cover completely the theory over complex numbers which is Character Theory. A large number of worked out examples are the main feature of these notes. The prerequisite for this note is basic group theory and linear algebra.
This chapter is on the representation theory of finite groups. For further details, we refer the reader to [23]. Throughout, G is a finite group of cardinality n and K …
This chapter is on the representation theory of finite groups. For further details, we refer the reader to [23]. Throughout, G is a finite group of cardinality n and K is algebraically closed of characteristic 0, or of characteristic p where p does not divide n.
Embeddings, geometries and representations - connections and computations, M.K. Bardoe Infinite dimensional modules for a finite group, D.J. Benson Degrees and diagrams of integral table algebras, H.I. Blau Canonical induction …
Embeddings, geometries and representations - connections and computations, M.K. Bardoe Infinite dimensional modules for a finite group, D.J. Benson Degrees and diagrams of integral table algebras, H.I. Blau Canonical induction formulae and the defect of a character, R. Boltje Counting characters in blocks, 2.9, E.C. Dade The defect groups of a clique, H. Ellers Representations of GLn(K) and symmetric groups, K. Erdmann On extended block induction and Brauer's third main theorem, G. Huang On blocks and source algebras for the double covers of the symmetric groups, R. Kessar A survey on the local structure of Morita and Rickard equivalences between Brauer blocks, L. Puig Some open conjectures on representation theory, G.R. Robinson Are all groups finite? L.L. Scott Locally finite varieties of groups and representations of finite groups, S.A. Syskin.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
This chapter is on the representation theory of finite groups. For further details, we refer the reader to [17]. Throughout, G is a finite group of cardinality n and K …
This chapter is on the representation theory of finite groups. For further details, we refer the reader to [17]. Throughout, G is a finite group of cardinality n and K is algebraically closed of characteristic 0, or of characteristic p where p does not divide n.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
We give a counterexample to the most optimistic analogue (due to Kleshchev and Ram) of the James conjecture for Khovanov-Lauda-Rouquier algebras associated to simply-laced Dynkin diagrams. The first counterexample occurs …
We give a counterexample to the most optimistic analogue (due to Kleshchev and Ram) of the James conjecture for Khovanov-Lauda-Rouquier algebras associated to simply-laced Dynkin diagrams. The first counterexample occurs in type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A 5"> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>5</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">A_5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals 2"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p = 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and involves the same singularity used by Kashiwara and Saito to show the reducibility of the characteristic variety of an intersection cohomology <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-module on a quiver variety. Using recent results of Polo one can give counterexamples in type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in all characteristics.
Given a stratified variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with strata satisfying a cohomological parity-vanishing condition, we define and show the uniqueness …
Given a stratified variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with strata satisfying a cohomological parity-vanishing condition, we define and show the uniqueness of “parity sheaves,” which are objects in the constructible derived category of sheaves with coefficients in an arbitrary field or complete discrete valuation ring. This construction depends on the choice of a parity function on the strata. If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> admits a resolution also satisfying a parity condition, then the direct image of the constant sheaf decomposes as a direct sum of parity sheaves, and the multiplicities of the indecomposable summands are encoded in certain refined intersection forms appearing in the work of de Cataldo and Migliorini. We give a criterion for the Decomposition Theorem to hold in the semi-small case. Our framework applies to many stratified varieties arising in representation theory such as generalised flag varieties, toric varieties, and nilpotent cones. Moreover, parity sheaves often correspond to interesting objects in representation theory. For example, on flag varieties we recover in a unified way several well-known complexes of sheaves. For one choice of parity function we obtain the indecomposable tilting perverse sheaves. For another, when using coefficients of characteristic zero, we recover the intersection cohomology sheaves and in arbitrary characteristic the special sheaves of Soergel, which are used by Fiebig in his proof of Lusztig’s conjecture.
We define two related invariants for a d-dimensional local ring (R,𝔪,k) called syzygy and differential symmetric signature by looking at the maximal free splitting of reflexive symmetric powers of two …
We define two related invariants for a d-dimensional local ring (R,𝔪,k) called syzygy and differential symmetric signature by looking at the maximal free splitting of reflexive symmetric powers of two modules: the top-dimensional syzygy module SyzRd(k) of the residue field and the module of Kähler differentials ΩR∕k of R over k. We compute these invariants for two-dimensional ADE singularities obtaining 1∕|G|, where |G| is the order of the acting group, and for cones over elliptic curves obtaining 0 for the differential symmetric signature. These values coincide with the F-signature of such rings in positive characteristic.
Following Lluis Puig we give a presentation of the theory of $p$-permutation modules (also called "trivial source modules") by a systematic use of the generalized Brauer morphism.
Following Lluis Puig we give a presentation of the theory of $p$-permutation modules (also called "trivial source modules") by a systematic use of the generalized Brauer morphism.
We show that the splendid Rickard complexes for blocks with Klein four defect groups constructed by Rickard and Linckelmann descend to non-split fields. As a corollary, Navarro's refinement of the …
We show that the splendid Rickard complexes for blocks with Klein four defect groups constructed by Rickard and Linckelmann descend to non-split fields. As a corollary, Navarro's refinement of the Alperin-McKay conjecture holds for blocks with a Klein four defect group. We also prove that splendid Morita equivalences between blocks and their Brauer correspondents (if exist) descend to non-split situations.
Journal Article Extending Brauer's Height Zero Conjecture to Blocks with Nonabelian Defect Groups Get access Charles W. Eaton, Charles W. Eaton 1School of Mathematics, Alan Turing Building, University of Manchester, …
Journal Article Extending Brauer's Height Zero Conjecture to Blocks with Nonabelian Defect Groups Get access Charles W. Eaton, Charles W. Eaton 1School of Mathematics, Alan Turing Building, University of Manchester, Oxford Road, Manchester M13 9PL, UK Correspondence to be sent to: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Alexander Moretó Alexander Moretó 2Departamento de Algebra, Facultad de Matematicas, Universidad de Valencia, 46100 Burjassot (Valencia), Spain Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2014, Issue 20, 2014, Pages 5581–5601, https://doi.org/10.1093/imrn/rnt131 Published: 03 July 2013 Article history Received: 07 March 2013 Revision received: 29 May 2013 Accepted: 04 June 2013 Published: 03 July 2013
Click to increase image sizeClick to decrease image size *We would like to thank the Institut für Experimentelle Mathematik in Essen for its hospitality and financial assitance during our stay …
Click to increase image sizeClick to decrease image size *We would like to thank the Institut für Experimentelle Mathematik in Essen for its hospitality and financial assitance during our stay in March 1993 when this research was undertaken. Notes *We would like to thank the Institut für Experimentelle Mathematik in Essen for its hospitality and financial assitance during our stay in March 1993 when this research was undertaken.
Abstract In this paper, the Brauer trees are completed for the sporadic simple Lyons group Ly in characteristics 37 and 67. The results are obtained using tools from computational representation …
Abstract In this paper, the Brauer trees are completed for the sporadic simple Lyons group Ly in characteristics 37 and 67. The results are obtained using tools from computational representation theory—in particular, a new condensation technique—and with the assistance of the computer algebra systems MeatAxe and GAP.
In this paper, we characterize the finite solvable groups with non-complete character degree graphs by proving the following theorem, which generalizes a conjecture by Huppert. Suppose that G is a …
In this paper, we characterize the finite solvable groups with non-complete character degree graphs by proving the following theorem, which generalizes a conjecture by Huppert. Suppose that G is a finite solvable group and p is a prime number dividing the degree of some irreducible character of G. If there is another such prime number q such that pq does not divide the degree of any irreducible character of G, then both p-length ℓ p (G) and q-length ℓ q (G) of G are at most two, and ℓ p (G)+ ℓ q (G)=4 if and only if pq=6 with Q G /Z φ (Q G )≅ 3 2 :GL(2,3), where Q G is generated by all Sylow 2-subgroups of G and Z φ (G) is a normal nilpotent subgroup of G. Moreover, the bounds are best possible.
Abstract. Except for blocks with a cyclic or Klein four defect group, it is not known in general whether the Morita equivalence class of a block algebra over a field …
Abstract. Except for blocks with a cyclic or Klein four defect group, it is not known in general whether the Morita equivalence class of a block algebra over a field of prime characteristic determines that of the corresponding block algebra over a p -adic ring. We prove this to be the case when the defect group is quaternion of order 8 and the block algebra over an algebraically closed field k of characteristic 2 is Morita equivalent to kà 4 . The main ingredients are Erdmann's classification of tame blocks [ 6 ] and work of Cabanes and Picaronny [ 4, 5 ] on perfect isometries between tame blocks.
Abstract We determine the character table of the endomorphism ring of the permutation module associated with the multiplicity-free action of the sporadic simple Baby Monster group B on its conjugacy …
Abstract We determine the character table of the endomorphism ring of the permutation module associated with the multiplicity-free action of the sporadic simple Baby Monster group B on its conjugacy class 2B, where the centraliser of a 2B-element is a maximal subgroup of shape 2 1+22 .Co 2 . This is one of the first applications of a new general computational technique to enumerate big orbits.
Since all finite simple groups have been classified [], it is a natural question whether the major conjectures in modular representation theory are consequences of this important and deep classification …
Since all finite simple groups have been classified [], it is a natural question whether the major conjectures in modular representation theory are consequences of this important and deep classification theorem. In this article, a survey is given about the progress which has been achieved on the famous open conjectures of J. Alperin [] and R. Brauer [] during the last decade. It turns out that all these problems have rather complete affirmative answers for almost all infinite series of finite groups like the symmetric, classical or exceptional groups of Lie type G for which there is a good parametrization of the irreducible characters χ of G into p-blocks B, where p is a prime divisor of the order of G. Using techniques from Clifford theory as described in Berger [] and chapter 10 of Feit [] it is often possible to reduce the proof of a general conjecture to the case of the automorphism groups of all the covering groups of a finite simple group. In particular, such reduction theorems exist for one direction of Brauer's height zero conjecture and for Alperin's weight conjecture as has been shown by Berger and Knörr [] and Dade [], respectively.
We first prove the conjecture mentioned by Leonard K. Jones in his thesis. By applying this conjecture, we obtain that the vertex of an indecomposable ${\mathcal {H}_F}$-module is an $l$-parabolic …
We first prove the conjecture mentioned by Leonard K. Jones in his thesis. By applying this conjecture, we obtain that the vertex of an indecomposable ${\mathcal {H}_F}$-module is an $l$-parabolic subgroup. Finally, we establish the Green correspondence for the representations of Hecke algebras of type ${A_{r - 1}}$.
R. Brauer not only laid the foundations of modular representation theory of finite groups, he also raised a number of questions and made conjectures (see [1], [2] for instance) which …
R. Brauer not only laid the foundations of modular representation theory of finite groups, he also raised a number of questions and made conjectures (see [1], [2] for instance) which since then have attracted the interest of many people working in the field and continue to guide the research efforts to a good extent. One of these is known as the “Height zero conjecture”. It may be stated as follows: CONJECTURE. Let B be a p-block of the finite group G. All irreducible ordinary characters of G belonging to B are of height 0 if and only if a defect group of B is abelian .
Let $p$ be a prime and $G$ a subgroup of $GL_d(p)$. We define $G$ to be $p$-exceptional if it has order divisible by $p$, but all its orbits on vectors …
Let $p$ be a prime and $G$ a subgroup of $GL_d(p)$. We define $G$ to be $p$-exceptional if it has order divisible by $p$, but all its orbits on vectors have size coprime to $p$. We obtain a classification of $p$-exceptional linear groups. This has consequences for a well-known conjecture in representation theory, and also for a longstanding question concerning $\frac {1}{2}$-transitive linear groups (i.e. those having all orbits on nonzero vectors of equal length), classifying those of order divisible by $p$.
The main problem of representation theory of finite groups is to find proofs of several conjectures stating that certain global invariants of a finite group G can be computed locally.The …
The main problem of representation theory of finite groups is to find proofs of several conjectures stating that certain global invariants of a finite group G can be computed locally.The simplest of these conjectures is the "McKay conjecture" which asserts that the number of irreducible complex characters of G of degree not divisible by p is the same if computed in a p-Sylow normalizer of G.In this paper, we propose a much stronger version of this conjecture which deals with Galois automorphisms.In fact, the same idea can be applied to the celebrated Alperin and Dade conjectures.
In this paper we study finite $p$-solvable groups having irreducible complex characters $\chi \in \operatorname {Irr}(G)$ which take root of unity values on the $p$-singular elements of $G$.
In this paper we study finite $p$-solvable groups having irreducible complex characters $\chi \in \operatorname {Irr}(G)$ which take root of unity values on the $p$-singular elements of $G$.