scholarly journals The mod-2 cohomology rings of symmetric groups

2012 ◽  
Vol 5 (1) ◽  
pp. 169-198 ◽  
Author(s):  
Chad Giusti ◽  
Paolo Salvatore ◽  
Dev Sinha
Keyword(s):  
Symmetric Groups ◽  
Cohomology Rings

scholarly journals Mod-two cohomology rings of alternating groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Chad Giusti ◽  
Dev Sinha
Keyword(s):  
Individual Component ◽  
Symmetric Groups ◽  
Ring Structure ◽  
Steenrod Algebra ◽  
Transfer Product ◽  
Additive Structure ◽  
New Techniques ◽  
Alternating Groups ◽  
Cohomology Rings ◽  
Abelian Subgroups

AbstractWe calculate the direct sum of the mod-two cohomology of all alternating groups, with both cup and transfer product structures, which in particular determines the additive structure and ring structure of the cohomology of individual groups. We show that there are no nilpotent elements in the cohomology rings of individual alternating groups. We calculate the action of the Steenrod algebra and discuss individual component rings. A range of techniques are developed, including an almost Hopf ring structure associated to the embeddings of products of alternating groups and Fox–Neuwirth resolutions, which are new techniques. We also extend understanding of the Gysin sequence relating the cohomology of alternating groups to that of symmetric groups and calculation of restriction to elementary abelian subgroups.

scholarly journals Dominant and global dimension of blocks of quantised Schur algebras

2021 ◽  
Author(s):  
Ming Fang ◽  
Wei Hu ◽  
Steffen Koenig
Keyword(s):  
Hecke Algebras ◽  
Symmetric Groups ◽  
Global Dimension ◽  
Group Algebras ◽  
Dominant Dimension ◽  
Schur Algebras ◽  
Homological Dimensions ◽  
Weyl Duality

AbstractGroup algebras of symmetric groups and their Hecke algebras are in Schur-Weyl duality with classical and quantised Schur algebras, respectively. Two homological dimensions, the dominant dimension and the global dimension, of the indecomposable summands (blocks) of these Schur algebras S(n, r) and $$S_q(n,r)$$ S q ( n , r ) with $$n \geqslant r$$ n ⩾ r are determined explicitly, using a result on derived invariance in Fang, Hu and Koenig (J Reine Angew Math 770:59–85, 2021).

scholarly journals Irreducible restrictions of representations of symmetric groups in small characteristics: reduction theorems

2018 ◽  
Vol 293 (1-2) ◽  
pp. 677-723 ◽  
Author(s):  
Alexander Kleshchev ◽  
Lucia Morotti ◽  
Pham Huu Tiep
Keyword(s):  
Symmetric Groups
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