Graded representation theory

Let be a group and let be a graded representation of , i.e. a functor from to the category of graded vector spaces with each piece finite-dimensional. Thus acts on each graded piece individually, each of which is an ordinary finite-dimensional representation. We want to define a character associated to a graded representation, but if [...]

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Set-multiset duality and supervector spaces

Recall that the elementary symmetric functions generate the ring of symmetric functions as a module over any commutative ring . A corollary of this result, although I didn’t state it explicitly, is that the elementary symmetric functions are algebraically independent, hence any ring homomorphism from the symmetric functions is determined freely by the images of [...]

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Introduction to symmetric functions

The theory of symmetric functions, which generalizes some ideas that came up in the previous discussion of Polya theory, can be motivated by thinking about polynomial functions of the roots of a monic polynomial . Problems on high school competitions often ask for the sum of the squares or the cubes of the roots of [...]

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Some examples of graded algebras

Often in mathematics we work in an algebra with the property that the “degree” of an element has a multiplicative property. For example, in a polynomial ring in variables we can define the degree of a monomial to be the vector of its degrees with respect to each variable, and the product of monomials corresponds [...]

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