The symmetric group: Algebraic formulas for some Sf 6j symbols and Sf⊇Sf1×Sf2 3jm symbols

R. W. Haase; R. Dirl

doi:10.1063/1.527163

The symmetric group: Algebraic formulas for some Sf 6j symbols and Sf⊇Sf1×Sf2 3jm symbols

Journal of Mathematical Physics ◽

10.1063/1.527163 ◽

1986 ◽

Vol 27 (4) ◽

pp. 900-913 ◽

Cited By ~ 5

Author(s):

R. W. Haase ◽

R. Dirl

Keyword(s):

Symmetric Group ◽

6J Symbols

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Representations of the Infinite Symmetric Group

10.1017/cbo9781316798577 ◽

2016 ◽

Cited By ~ 3

Author(s):

Alexei Borodin ◽

Grigori Olshanski

Keyword(s):

Symmetric Group ◽

Infinite Symmetric Group

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The Transition Matrix Between the Specht and 𝔰𝔩3 Web Bases is Unitriangular With Respect to Shadow Containment

International Mathematics Research Notices ◽

10.1093/imrn/rnaa290 ◽

2020 ◽

Author(s):

Heather M Russell ◽

Julianna Tymoczko

Keyword(s):

Symmetric Group ◽

Partial Order ◽

Transition Matrix ◽

Group Representation ◽

Jones Polynomial ◽

Basis Matrix ◽

Combinatorial Structures ◽

Quantum Invariants ◽

One Dimensional ◽

Link Diagrams

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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