Periodicity in the cohomology of symmetric groups via divided powers

In [1] Atiyah described how to use the complex representations of the symmetric group, Sn, to define and investigate operations in complex topological K-theory. In this paper operations for more general Grothendieck groups are described in terms of the integral representations of Sn using the representations directly without passing to the dual as Atiyah did. The principal tool, which is proved in the first section, is the theorem that the direct sum of the Grothendieck groups of finite integral representations of Sn form a bialgebra isomorphic to a polynomial ring with a sequence of divided powers. A consequence of this theorem is that the only operations that can be constructed from the symmetric groups will be polynomials in the symmetric powers.

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Climate Policy in a System of Divided Powers: Dealing with Carbon Leakage and Regulatory Linkage

SSRN Electronic Journal ◽

10.2139/ssrn.2174024 ◽

2012 ◽

Author(s):

Daniel A. Farber

Keyword(s):

Climate Policy ◽

Carbon Leakage ◽

Divided Powers

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Dangerous Liaisons

10.1093/oso/9780198732174.003.0005 ◽

2017 ◽

Author(s):

András Sajó ◽

Renáta Uitz

Keyword(s):

Executive Branch ◽

Individual Liberty ◽

Checks And Balances ◽

Functional Distribution ◽

Governmental Functions ◽

Private Actors ◽

Divided Powers

This chapter examines the idea of separating distinct governmental functions into at least three branches (horizontal separation) as a means to safeguard individual liberty. The three branches of government have different functions: the legislature legislates, the executive branch executes the laws, and the judiciary administers justice. This corresponds to the functional distribution of essential governmental tasks and competences. The chapter explores how governments based on separated (or at least divided) powers work, in a perpetual balancing exercise as a result of the operation of checks and balances. Finally, it discusses independent agencies that are now routinely added to the old constitutional mix of powers and the problem of outsourcing public powers to private actors.

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Collapsed adjacency diagrams and matrices of commuting graph, 𝒞 (G, X ) for some symmetric groups, Sym(n)

10.1063/5.0053458 ◽

2021 ◽

Author(s):

Athirah Nawawi ◽

Peter J. Rowley

Keyword(s):

Symmetric Groups ◽

Commuting Graph

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Dominant and global dimension of blocks of quantised Schur algebras

Mathematische Zeitschrift ◽

10.1007/s00209-021-02792-w ◽

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).

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