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تسجيل مستخدم جديدThe transition matrix between the Specht and $mathfrak{sl}_3$ web bases is unitriangular with respect to shadow containment
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Webs are planar graphs with boundary that describe morphisms in a diagrammatic representation category for $mathfrak{sl}_k$. They are studied extensively by knot theorists because braiding maps provide a categorical way to express link diagrams in terms of webs, producing quantum invariants like the well-known Jones polynomial. One important question in representation theory is to identify the relationships between different bases; coefficients in the change-of-basis matrix often describe combinatorial, algebraic, or geometric quantities (like, e.g., Kazhdan-Lusztig polynomials). By flattening the braiding maps, webs can also be viewed as the basis elements of a symmetric-group representation. In this paper, we define two new combinatorial structures for webs: band diagrams and their one-dimensional projections, shadows, that measure depths of regions inside the web. As an application, we resolve an open conjecture that the change-of-basis between the so-called Specht basis and web basis of this symmetric-group representation is unitriangular for $mathfrak{sl}_3$-webs. We do this using band diagrams and shadows to construct a new partial order on webs that is a refinement of the usual partial order. In fact, we prove that for $mathfrak{sl}_2$-webs, our new partial order coincides with the tableau partial order on webs studied by the authors and others. We also prove that though the new partial order for $mathfrak{sl}_3$-webs is a refinement of the previously-studied tableau order, the two partial orders do not agree for $mathfrak{sl}_3$.
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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 alg
ebraic 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.
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