Introduction E9F5FC

Questions FFFFC0

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Readings

• Elementary symmetric polynomial
• Exterior algebra
• Schur functor

Facts

In representation theory, the exterior algebra is one of the two fundamental Schur functors on the category of vector spaces, the other being the symmetric algebra. Together, these constructions are used to generate the irreducible representations of the general linear group; see fundamental representation. (Exterior algebra)

Elementary symmetric polynomials

There are {$2^n$} terms in all for the elementary symmetric polynomials on {$n$} variables. They express the ways of choosing {$k$} out of {$n$} items for all values of {$k$}. When {$k>n$} then there are no ways.

In other words, given a simplex, these are all the ways of choosing a subsimplex, thus tearing it down, as opposed to building it up.

These are inclusion morphisms and they can be composed.

Products of polynomials

A product of polynomials would yield a pair of possibly overlapping subsets.

Determinant (Elementary symmetric function of eigenvalues)

The determinant is a quintessential geometric value, an n-dimensional volume.