scholarly journals Orbifolds and finite group representations

2001 ◽  
Vol 26 (11) ◽  
pp. 649-669
Author(s):  
Li Chiang ◽  
Shi-Shyr Roan
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Representation Theory ◽  
Hilbert Scheme ◽  
Group Representation ◽  
Quotient Singularity ◽  
Group Representations ◽  
Group Representation Theory ◽  
Crepant Resolutions

We present our recent understanding on resolutions of Gorenstein orbifolds, which involves the finite group representation theory. We concern only the quotient singularity of hypersurface type. The abelian groupAr(n)forA-type hypersurface quotient singularity of dimensionnis introduced. Forn=4, the structure of Hilbert scheme of group orbits and crepant resolutions ofAr(4)-singularity are obtained. The flop procedure of4-folds is explicitly constructed through the process.

On Extending Projectives of Finite Group-Graded Algebras

1991 ◽  
Vol 34 (2) ◽  
pp. 224-228
Author(s):  
Morton E. Harris
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Group Representation ◽  
Sufficient Conditions ◽  
Projective Cover ◽  
Graded Algebras ◽  
Finite Dimensional ◽  
Completely Reducible ◽  
A Finite Group ◽  
Necessary And Sufficient

AbstractLet G be a finite group, let k be a field and let R be a finite dimensional fully G-graded k-algebra. Also let L be a completely reducible R-module and let P be a projective cover of R. We give necessary and sufficient conditions for P|R1 to be a projective cover of L|R1 in Mod (R1). In particular, this happens if and only if L is R1-projective. Some consequences in finite group representation theory are deduced.

scholarly journals Stratifications and Mackey Functors II: Globally Defined Mackey Functors

2010 ◽  
Vol 6 (1) ◽  
pp. 99-170 ◽  
Author(s):  
Peter Webb
Keyword(s):  
Representation Theory ◽  
Structural Properties ◽  
Characteristic Zero ◽  
Group Representation ◽  
Cartan Matrix ◽  
Weight Category ◽  
Group Representation Theory ◽  
Stratification Theory ◽  
Highest Weight ◽  
Mackey Functors

AbstractWe describe structural properties of globally defined Mackey functors related to the stratification theory of algebras. We show that over a field of characteristic zero they form a highest weight category and we also determine precisely when this category is semisimple. This approach is used to show that the Cartan matrix is often symmetric and non-singular, and we are able to compute finite parts of it in some instances. We also develop a theory of vertices of globally defined Mackey functors in the spirit of group representation theory, as well as giving information about extensions between simple functors.

scholarly journals On hypersurface quotient singularities of dimension 4

2004 ◽  
Vol 2004 (48) ◽  
pp. 2547-2581
Author(s):  
Li Chiang ◽  
Shi-Shyr Roan
Keyword(s):  
Abelian Group ◽  
Hilbert Scheme ◽  
Alternating Group ◽  
Detailed Structure ◽  
Special Group ◽  
Standard Representation ◽  
Quotient Singularities ◽  
Crepant Resolutions

We consider geometrical problems on Gorenstein hypersurface orbifolds of dimensionn≥4through the theory of Hilbert scheme of group orbits. For a linear special groupGacting onℂn, we study theG-Hilbert schemeHilbG(ℂn)and crepant resolutions ofℂn/GforGtheA-type abelian groupAr(n). Forn=4, we obtain the explicit structure ofHilbAr(4)(ℂ4). The crepant resolutions ofℂ4/Ar(4)are constructed through their relation withHilbAr(4)(ℂ4), and the connections between these crepant resolutions are found by the “flop” procedure of 4-folds. We also make some primitive discussion onHilbG(ℂn)forGthe alternating group𝔄n+1of degreen+1with the standard representation onℂn; the detailed structure ofHilb𝔄4(ℂ3)is explicitly constructed.

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