English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The first half of this paper is largely expository, wherein we present asystematic combinatorial approach to the theory of polynomial (semi)invariantsand multilinear invariants of several vectors and covectors, for the classicalgroups. This culminates in a graphical description of graded linear bases. Byapplying well-known results of lattice path combinatorics to Weyl's fundamentaltheorems of classical invariant theory, we write down Stanley decompositionsand Hilbert-Poincare series in terms of families of non-intersecting latticepaths, enumerated with respect to certain corners.  In the second half of the paper, we revisit the (semi)invariants in the firsthalf as a special case of a much broader phenomenon. On one hand, polynomialinvariants of a group $H$ can be generalized to modules of covariants, i.e.,$H$-equivariant polynomial functions between $H$-modules. On the other hand,from the perspective of Roger Howe's theory of dual pairs, these modules ofcovariants can be viewed as infinite-dimensional simple $(\mathfrak{g},K)$-modules. This suggests an expanded program in which our goal is to applycombinatorial techniques involving lattice paths in order to write down Hilbertseries for arbitrary unitarizable highest-weight $(\mathfrak{g},K)$-modules. Asa preview of future work in this program, we present examples showing howmodules of covariants -- even those which are not Cohen-Macaulay, and thereforewhich we would not expect to be combinatorially nice -- can be decomposed interms of lattice paths. We also extend these methods beyond the classicalgroups.

Stanley decompositions of rings of invariants and certain highest weight Harish-Chandra modules