# linear algebra – How are symmetric functions related to Koszul duality?

Staying within the world of linear algebra, we have the following two “dualities” between exterior powers and symmetric powers.

The first is that of Kozsul duality, so these two graded algebras $$Lambda^*(V)$$ and $$S^*(hat{V})$$ are the Ext’s of each other, a homological duality.

The second is the duality between elementary symmetric functions and homogenous symmetric functions, which can be interpreted as a canonical, non-obvious involution on the ring of infinite symmetric polynomials. This second duality has easily visible consequences on the level of power series of many combinatorial objects, given by inversion and sending $$tmapsto -t$$, for instance in the Poincare series of these two graded algebras.

Now while I can draw parallels between these two specific duality operations, my question is about the nature of this duality in general. In particular, I am (vaguely) aware that Koszul duality goes much higher, and fits into a more general picture, and on the symmetric functions side, there is much more structure present than just the elementary and homogeneous symmetric functions.

So then, does this extra structure of symmetric functions (say, being the free $$lambda$$ ring on one generator) have an analogue in the more general framework of Kozsul duality?

As a specific question, does anyone know of an example of either Schur symmetric function combinatorics or the Frobenius lifts/Lambda structure in some (perhaps generalised) instance of Koszul duality?