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.
Highlights
In this paper we study two closely-related pairs of objects: 1) graphs equipped with an edge-labeling that represents the action of certain permutations, and 2) natural bases for representations of the symmetric group
Using properties of the graphs, we prove that the bases are related by a transition matrix that is upper-triangular with ones along the diagonal
The vertices are standard Young tableaux of shape (n, n): in other words they are ways to fill a 2 × n grid with the integers 1, 2, . . . , 2n so that numbers increase left-to-right in each row and top-to-bottom in each column
Summary
In this paper we study two closely-related pairs of objects: 1) graphs equipped with an edge-labeling that represents the action of certain permutations, and 2) natural bases for representations of the symmetric group. Standard Young tableaux are fundamental objects not just in combinatorics and in representation theory and geometry (see, e.g., Fulton’s book for an overview [9]) They arise in our context as a geometrically-natural characterization of the cells in a topological decomposition of a family of varieties called Springer fibers [10, 28, 29, 33]. The perspective of webs gives us new information about changeof-basis coefficients in the sl case and provides a foundation for studying more general web and Specht bases To relate these bases, we construct a web graph and a tableau graph with vertices of webs and standard Young tableaux respectively. The authors gratefully acknowledge helpful comments from Sabin Cautis, Mikhail Khovanov, Greg Kuperberg, Brendon Rhoades, Anne Schilling, and John Stembridge
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