Generic and $q$-Rational Representation Theory

Cline, Edward; Parshall, Brian; Scott, Leonard L.

doi:10.2977/prims/1195144189

Cited by 22 publications

(27 citation statements)

References 41 publications

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“…We note that this was already proven in [8] with less effort but our proof contains already some key arguments to be used in the proof of the main theorem.…”

Section: Corollary If Splits and If Is Noetherian Thenmentioning

confidence: 99%

“…What we have proven so far can be summarized as follows. Cline-Parshall-Scott [8] also study several properties on classes of algebras and show that they are generic (openness is not considered). The following example from [8] illustrates that even "honest algebraic" properties might not be generic in general and so cannot be described by a discriminant as above.…”

Section: Theorem If Is Noetherian Andmentioning

confidence: 99%

“…The ultimate aim is to understand the representation theory (or some specific part of it) of all fibers of . The correct approach to this problem-following the philosophy of Grothendieck and what Cline-Parshall-Scott [8] coined generic representation theory-is to understand how the representation theory of (p) varies with p. To this end, it suffices to understand how the representation theory of a special fiber (p) relates to the one of the only distinguished fiber of , the generic fiber (0) = , where = ⊗ . The hope is that this relation is generically strong so that once we understand the generic fiber , we understand Zariski almost all fibers.…”

Section: Introductionmentioning

confidence: 99%

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Decomposition Matrices are Generically Trivial

Thiel¹

2015

Int Math Res Notices

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ABSTRACT. We establish a genericity property in the representation theory of a flat family of finite-dimensional algebras in the sense of Cline-Parshal-Scott. More precisely, we show that the decomposition matrices as introduced by Geck and Rouquier of an algebra which is free of finite dimension over a noetherian integral domain and which splits over the fraction field of this ring are generically trivial, i.e., they are trivial in an open neighborhood of the generic point of the spectrum of the base ring. This generalizes a classical result by Brauer in modular representation theory of finite groups. We furthermore show that this is true precisely on an open set in case all fibers of the algebra split. In this way we get a stratification of the base scheme such that decomposition maps are trivial on each stratum. Moreover, this defines a new discriminant of such algebras which generalizes Schur elements of simple modules for symmetric split semisimple algebras. We provide some extensions to the theory of decomposition maps allowing us to work without the usual normality assumption on the base ring.

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“…We note that this was already proven in [8] with less effort but our proof contains already some key arguments to be used in the proof of the main theorem.…”

Section: Corollary If Splits and If Is Noetherian Thenmentioning

confidence: 99%

Section: Theorem If Is Noetherian Andmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Decomposition Matrices are Generically Trivial

Thiel¹

2015

Int Math Res Notices

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“…Usually, we shall fix an A-algebra k which is a field, and which as a field is C, and we shall consider H(n, d) = H A (n, d) ⊗ A k. The semisimple generic structure of H(n, d) over C is well known, through that of the specialisation to the group algebra of the group Z d S n (confer [13,33,35] as in [2]). We recall it briefly.…”

Section: Figure 1: Computation Of the Polynomials N A (B)mentioning

confidence: 99%

“…I1 A tower of unital algebras A n ⊂ A n+1 over a ring R with indeterminates, and a multiplicity-free [69] semisimple specialisation [13] (split, and we shall consider only characteristic 0 here).…”

Section: Figure 1: Computation Of the Polynomials N A (B)mentioning

confidence: 99%

Generalized Blob Algebras and Alcove Geometry

Martin¹,

Woodcock

2003

LMS J. Comput. Math.

122

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A sequence of finite-dimensional quotients of affine Hecke algebras is studied. Each element of the sequence is constructed so as to have a weight space labelling scheme for Specht/standard modules. As in the weight space formalism of algebraic Lie theory, there is an action of an affine reflection group on this weight space that fixes the set of labelling weights. A linkage principle is proved in each case. Further, it is shown that the simplest non-trivial example may essentially be identified with the blob algebra (a physically motivated quasihereditary algebra whose representation theory is very well understood by Lie-theory-like methods). An extended role is hence proposed for Soergel's tilting algorithm, away from its algebraic Lie theory underpinning, in determining the simple content of standard modules for these algebras. This role is explicitly verified in the blob algebra case. A tensor space representation of the blob algebra is constructed, as a candidate for a full tilting module (subsequently proven to be so in a paper by Martin and Ryom-Hansen), further evidencing the extended utility of Lie-theoretic methods. Possible generalisations of this representation to other elements of the sequence are discussed.

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Linear and Nonlinear Group Actions, and the Newton Institute Program

Scott

1998

Algebraic Groups and Their Representations

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One of the simplest and most useful notions in mathematics is that of a group action: if G is a group and X is a nonempty set, then an action of G on X (or a G-set structure on X) consists of a multiplication operation G x X -+ X, with the image of a pair (g, x) written as, say, gx, with the following axioms satisfied:(1) 1x = x for all x E X (here 1 EGis the identity element of G)j (2) (gh)x = g(hx) for all g, hE G and all x E X.Equivalently, a group action determines, and is determined by, a homomorphism 9 1-+ [x 1-+ gx] of G into the group of all bijective maps of X onto itself (the 'symmetric' group of all permutations of X). If X has the additional structure of a linear space over a field, we might impose an additional axiom that the mappings [x ~ gx] all be linear. In this case, we say the action of G is linear, or that it defines a linear representation of G, and we call X a module for Gover k. In the linear or nonlinear case, if G and X have some topological structure, we often require that the map G x X ----t X be continuous, or in the algebraic geometry case, a morphism of algebraic varieties or schemes. A main theme of the Newton Institute program! on the representation theory of algebraic and related finite groups, of which this conference volume is a part, may then be stated as follows: even in the simplest case where G is a finite group, and X is a finite set with no linear space or algebraic variety structure, there are, nevertheless, deep connections of these nonlinear actions with linear actions, and of the discrete theory of finite groups and their actions with the theory of algebraic groups and other groups and algebras arising in continuous Lie theory. I hope to explain these connections IThe author would like to thank the NSF for its support and the Newton Institute for its hospitality.

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Generic and $q$-Rational Representation Theory

Cited by 22 publications

References 41 publications

Decomposition Matrices are Generically Trivial

Decomposition Matrices are Generically Trivial

Generalized Blob Algebras and Alcove Geometry

Linear and Nonlinear Group Actions, and the Newton Institute Program

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