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Register a new userThe 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 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.
We continue investigating the generalisations of geometrical statistical models introduced in [13], in the form of models of webs on the hexagonal lattice H having a U_q(sl_n) quantum group symmetry. We focus here on the n=3 case of cubic webs, based on the Kuperberg A_2 spider, and illustrate its properties by comparisons with the well-known dilute loop model (the n=2 case) throughout. A local vertex-model reformulation is exhibited, analogous to the correspondence between the loop model and a three-state vertex model. The n=3 representation uses seven states per link of H, displays explicitly the geometrical content of the webs and their U_q(sl_3) symmetry, and permits us to study the model on a cylinder via a local transfer matrix. A numerical study of the central charge reveals that for each q $in mathbb{C}$ in the critical regime, |q|=1, the web model possesses a dense and a dilute critical point, just like its loop model counterpart. In the dense $q=-e^{i pi/4}$ case, the n=3 webs can be identified with spin interfaces of the critical three-state Potts model defined on the triangular lattice dual to H. We also provide another mapping to a $mathbb{Z}_3$ spin model on H itself, using a high-temperature expansion. We then discuss the sector structure of the transfer matrix, for generic q, and its relation to defect configurations in both the strip and the cylinder geometries. These defects define the finite-size precursors of electromagnetic operators. This discussion paves the road for a Coulomb gas description of the conformal properties of defect webs, which will form the object of a subsequent paper. Finally, we identify the fractal dimension of critical webs in the $q=-e^{i pi/3}$ case, which is the n=3 analogue of the polymer limit in the loop model.
The $q$-analog of Kostants weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the $q$-analog of Kostants partition function. This formula, when evaluated at $q=1$, gives the multiplicity of a weight in a highest weight representation of a simple Lie algebra. In this paper, we consider the Lie algebra $mathfrak{sl}_4(mathbb{C})$ and give closed formulas for the $q$-analog of Kostants weight multiplicity. This formula depends on the following two sets of results. First, we present closed formulas for the $q$-analog of Kostants partition function by counting restricted colored integer partitions. These formulas, when evaluated at $q=1$, recover results of De Loera and Sturmfels. Second, we describe and enumerate the Weyl alternation sets, which consist of the elements of the Weyl group that contribute nontrivially to Kostants weight multiplicity formula. From this, we introduce Weyl alternation diagrams on the root lattice of $mathfrak{sl}_4(mathbb{C})$, which are associated to the Weyl alternation sets. This work answers a question posed in 2019 by Harris, Loving, Ramirez, Rennie, Rojas Kirby, Torres Davila, and Ulysse.
The Witt group of nondegenerate braided fusion categories $mathcal{W}$ contains a subgroup $mathcal{W}_text{un}$ consisting of Witt equivalence classes of pseudo-unitary nondegenerate braided fusion categories. For each finite-dimensional simple Lie algebra $mathfrak{g}$ and positive integer $k$ there exists a pseudo-unitary category $mathcal{C}(mathfrak{g},k)$ consisting of highest weight integerable $hat{g}$-modules of level $k$ where $hat{mathfrak{g}}$ is the corresponding affine Lie algebra. Relations between the classes $[mathcal{C}(mathfrak{sl}_2,k)]$, $kgeq1$ have been completely described in the work of Davydov, Nikshych, and Ostrik. Here we give a complete classification of relations between the classes $[mathcal{C}(mathfrak{sl}_3,k)]$, $kgeq1$ with a view toward extending these methods to arbitrary simple finite dimensional Lie algebras $mathfrak{g}$ and positive integer levels $k$.
Recently, Gnutzmann and Smilansky presented a formula for the bond scattering matrix of a graph with respect to a Hermitian matrix. We present another proof for this Gnutzmann and Smilanskys formula by a technique used in the zeta function of a graph. Furthermore, we generalize Gnutzmann and Smilanskys formula to a regular covering of a graph. Finally, we define an $L$-fuction of a graph, and present a determinant expression. As a corollary, we express the generalization of Gnutzmann and Smilanskys formula to a regular covering of a graph by using its $L$-functions.
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