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Abstract 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 (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, which 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 ([ 33] and [ 29].) 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 [ 12, 17, 29, 33]. 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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Products of symmetric group characters
In \cite{OZ}, the authors introduced a new basis of the ring of symmetric functions which evaluate to the irreducible characters of the symmetric group at roots of unity. The structure coefficients for this new basis are the stable Kronecker coefficients. In this paper we give combinatorial descriptions for several products. In addition, we identify some applications and instances where special cases of these products have occurred elsewhere in the mathematical literature.
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- Award ID(s):
- 1700058
- PAR ID:
- 10097210
- Date Published:
- Journal Name:
- Journal of combinatorial theory. Series A
- Volume:
- 165
- ISSN:
- 1096-0899
- Page Range / eLocation ID:
- 299-324
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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