The Transition Matrix between the Specht and Web Bases Is Unipotent with Additional Vanishing Entries

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.

The Polytabloid Basis Expands Positively Into the Web Basis

“…The purpose of this note is to establish a positivity relation between the polytabloid and web bases of the Catalan-dimensional irreducible S 2n -representation of shape (n, n). This proves a conjecture of Russell and Tymoczko [7,Conj. 5.8].…”

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We show that the transition matrix from the polytabloid basis to the web basis of the irreducible S2n-representation of shape (n, n) has nonnegative integer entries. This proves a conjecture of Russell and Tymoczko [7]. arXiv:1808.00445v1 [math.CO]

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“…Theorem 1.2 will follow if we can show that the sign appearing in Equation (2.11) is always positive. But this follows from the unitriangularity result Theorem 1.1 of Russell and Tymoczko [7] combined with Lemma 2.1.…”

Set partitions, fermions, and skein relations

“…The invariant subring C[X] SL 2 is generated by the minors {∆ ab : 1 ≤ a < b ≤ n} and the syzygy ideal of relations among these generators is generated by Kung and Rota [9] gave a detailed combinatorial study of this invariant ring which has seen representation-theoretic application (e.g. [16,20]). One might expect that the 3-term and 4term skein relations in Figure 1 correspond to syzygies among generators of some invariant ring R.…”

“…The skein action was introduced to give representation theoretic proofs of cyclic sieving results of Reiner-Stanton-White [14] and Pechenik [11] involving the rotational action of Z n on various sets of noncrossing partitions of [n]. Skein bases generalize the Kazhdan-Lusztig cellular and sl 2 -web bases (see [9,13,16,20]) of symmetric group irreducibles labeled by 2-row rectangles.…”