The symmetric functions catalog

#The Yang–Baxter equations

Let \(V\) be a vector space. A linear operator \(R\) on \(V\otimes V\) is said to be a solution to the Yang–Baxter equation if \[R^{12} \circ R^{23} \circ R^{12} = R^{23} \circ R^{12} \circ R^{23}\] when acting on \(V \times V \times V,\) where \(R^{ij}\) means that we act on components \(i\) and \(j.\) Stated differently, we wish that \[(R \times Id)\circ (Id \times R)\circ (R \times Id) = (Id \times R) \circ (R \times Id) \circ (Id \times R).\]

Another form is to have some operators, \(h_i,\) so that \[h_i(x)h_{i+1}(x+y)h_i(y) = h_{i+1}(y) h_{i}(x+y) h_{i+1}(x)\] See [FK96] for the connection with Schubert polynomials.

Yet another reference state the Yang–Baxter relation as \[R^{12} \circ R^{13} \circ R^{23} = R^{23} \circ R^{13} \circ R^{12}.\] Fonseca’s RAQIS 2012 notes use this convention.

Notions close to the Yang–Baxter equation are (integrable) vertex models, R-matrix, 5-vertex model, partition function.

It is interesting to show that symmetric functions are partition functions for some particular choice of a vertex model. This is closely related to Jacobi–Trudi identities.

#The 6 vertex model

B. Brubaker, D. Bump, and S. Friedberg construct a parametrized Yang–Baxter equation for six- and eight-vertex models and use it to prove a Hamel–King partition-function formula for Schur polynomials [BBF11]. Their model gives a direct bridge between strict Gelfand–Tsetlin patterns, Tokuyama’s formula, and six-vertex partition functions.

There is also a set-theoretic Yang–Baxter viewpoint on Young tableaux. V. Lebed shows that the plactic product on tableaux is determined by a braiding on columns, where the braiding is a set-theoretic solution to the Yang–Baxter equation [Leb20]. This gives another route from Yang–Baxter structures to the Schensted/plactic side of symmetric-function combinatorics.

For general background on the Yang–Baxter equation and quantum enveloping algebras, see [Ma93].

K. Motegi and T. Scrimshaw use an integrable vertex model for refined dual stable Grothendieck polynomials [MS25].

Bibliography

1. [BBF11]Ben Brubaker, Daniel Bump and Solomon Friedberg. Schur Polynomials and the Yang–Baxter equation. Communications in Mathematical Physics, 308(2):281–301, 2011.
   .bib

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   @article{BrubakerBumpFriedberg2011,
     author = {Ben Brubaker and Daniel Bump and Solomon Friedberg},
     title = {Schur {P}olynomials and the {Y}ang--{B}axter Equation},
     year = {2011},
     journal = {Communications in Mathematical Physics},
     volume = {308},
     number = {2},
     pages = {281--301},
     doi = {10.1007/s00220-011-1345-3},
     url2 = {https://doi.org/10.1007/s00220-011-1345-3},
     eprint = {0912.0911}
   }
   

2. [FK96]Sergey Fomin and Anatol N. Kirillov. The Yang–Baxter equation, symmetric functions, and Schubert polynomials. Discrete Math., 153(1-3):123–143, 1996.
   .bib

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   @article{FominKirillov1996YangBaxter,
   author={Sergey Fomin and Anatol N. Kirillov},
   title={The {Y}ang--{B}axter equation, symmetric functions, and {S}chubert polynomials},
   journal={Discrete Math.},
   year={1996},
   volume={153},
   number={1-3},
   pages={123--143}
   }
   

3. [Leb20]Victoria Lebed. Plactic monoids: A braided approach. Journal of Algebra, 564:325–352, 2020.
   .bib

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   @article{Lebed2020Plactic,
     author = {Victoria Lebed},
     title = {Plactic monoids: a braided approach},
     year = {2020},
     journal = {Journal of Algebra},
     volume = {564},
     pages = {325--352},
     doi = {10.1016/j.jalgebra.2020.08.010},
     url2 = {https://doi.org/10.1016/j.jalgebra.2020.08.010},
     eprint = {hal-01417747}
   }
   

4. [Ma93]Zhong-Qi Ma. Yang–baxter equation and quantum enveloping algebras. World Scientific, 1993.
   .bib

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   @book{Ma1993YangBaxter,
     author = {Zhong-Qi Ma},
     title = {Yang--Baxter Equation and Quantum Enveloping Algebras},
     year = {1993},
     publisher = {World Scientific},
     doi = {10.1142/2013},
     url2 = {https://doi.org/10.1142/2013}
   }
   

5. [MS25]Kohei Motegi and Travis Scrimshaw. Refined dual Grothendieck polynomials, integrability, and the Schur measure. Selecta Mathematica, 31(3), 2025.
   .bib

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   @article{MotegiScrimshaw2025,
     author = {Kohei Motegi and Travis Scrimshaw},
     title = {Refined dual {G}rothendieck polynomials, integrability, and the
       {S}chur measure},
     year = {2025},
     journal = {Selecta Mathematica},
     volume = {31},
     number = {3},
     doi = {10.1007/s00029-025-01041-w},
     eprint = {2012.15011}
   }