Document Type
Article
Publication Date
3-6-2019
Publication Title
International Mathematics Research Notices
Abstract
We compare two important bases of an irreducible representation of the symmetric group: the web basis and the Specht basis. The web basis has its roots in the Temperley-Lieb algebra and knot-theoretic considerations. The Specht basis is a classic algebraic and combinatorial construction of symmetric group representations which arises in this context through the geometry of varieties called Springer fibers. We describe a graph that encapsulates combinatorial relations between each of these bases, prove that there is a unique way (up to scaling) to map the Specht basis into the web representation, and use this to recover a result of Garsia-McLarnan that the transition matrix between the Specht and web bases is upper-triangular with ones along the diagonal. We then strengthen their result to prove vanishing of certain additional entries unless a nesting condition on webs is satisfied. In fact we conjecture that the entries of the transition matrix are nonnegative and are nonzero precisely when certain directed paths exist in the web graph.
Volume
2019
Issue
5
First Page
1479
Last Page
1502
DOI
10.1093/imrn/rnx164
ISSN
10737928
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Rights
Licensed to Smith College and distributed CC-BY under the Smith College Faculty Open Access Policy.
Recommended Citation
Russell, Heather M. and Tymoczko, Julianna S., "The Transition Matrix Between the Specht and Web Bases is Unipotent with Additional Vanishing Entries" (2019). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/102
Comments
Peer reviewed accepted manuscript.