Improve this page

#Rigged configurations

Rigged configurations were introduced by A. Kirillov and N. Reshetikhin [KR88]. One of the main applications is to compute Kostka–Foulkes polynomials efficiently.

We introduce the terminology following [DLT94].

Let \(\lambda\) and \(\mu\) be partitions of \(n.\) A matrix of type \((\lambda,\mu)\) is a matrix \(M=(m_{ij})\) with finitely many nonzero integer entries such that \[\sum_{j\geq 1} m_{ij} = \lambda_j \qquad \sum_{i\geq 1} m_{ij} = \mu'_j.\] We also associate matrices \(P\) and \(Q\) with \(M\) by \[P_{ij} = \sum_{k \leq j } \left( m_{i,k} - m_{i+1,k} \right) \qquad Q_{ij} = \sum_{k \geq i+1 } \left( m_{k,j} - m_{k,j+1} \right).\] The matrix \(M\) is admissible if and only if the entries in \(P\) and \(Q\) are nonnegative. The numbers \(P_{ij}\) for which \(Q_{ij}\gt 0\) are called the vacancy numbers of \(M.\)

#Configurations

An admissible matrix can be represented via a sequence of partitions, \(\nuvec = (\nu^0,\nu^1,\dotsc),\) such that \[(\nu^i)'_j = \sum_{k \geq i+1} m_{kj}.\] Note that \(\nu^0 = \mu\) and that \(\lambda_i = |\nu^i|-|\nu^{i-1}|\) for \(i \geq 1.\) Each entry \(Q_{ij}\) gives the multiplicity of occurrence of the value \(j\) as a part of the partition \(\nu^i.\)

A rigging of a configuration is defined as follows. For each \(i\) and each part size \(j\) appearing in \(\nu^i,\) an integer partition \(I^i_j\) is assigned, in such a way that the largest part of \(I^i_j\) does not exceed \(P_{ij},\) and the length of \(I^i_j\) does not exceed \(Q_{ij},\) which is also the number of parts of size \(j\) in \(\nu^i.\)

The parts in the partitions \(I^i_j\) are traditionally written in the leftmost box of the part in the Young diagram associated with \(\nu^i.\)

Example (Rigged configuration).

Consider the following matrix \(M,\) and the associated \(P\) and \(Q.\) The vacancy numbers in \(P\) have been marked in bold. \[M= \begin{pmatrix} 3 & 2 & 1 & -1 & 1 & 1 \\ 2 & 1 & 1 & 1 & 0 & 1 \\ 2 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \end{pmatrix}\] \[P= \begin{pmatrix} \mathbf{1} & 2 & \mathbf{2} & \mathbf{0} & 1 & \mathbf{1} \\ \mathbf{0} & 0 & \mathbf{0} & \mathbf{0} & \mathbf{0} & 1 \\ 1 & 1 & \mathbf{1} & 1 & \mathbf{0} & 0 \\ 0 & 0 & \mathbf{0} & 1 & 2 & 2 \\ 1 & 2 & 3 & 3 & 3 & 3 \\ \end{pmatrix} \; Q= \begin{pmatrix} 2 & 0 & 1 & 2 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{pmatrix}\] The type \((\lambda,\mu)\) of \(M\) is \((7 6 5 5 3, 6 6 3 3 3 2 1 1 1),\) and the associated \((\nu^i)_{i=0}^4\) is given by \[\nu^0= 663332111,\quad \nu^1 = 6 4 4 3 1 1 ,\quad \nu^2 = 5 4 3 1 ,\quad \nu^3= 53 ,\quad \nu^4 = 3.\] A rigged configuration for this \(\nuvec\) is given by the following decorated sequence of Young diagrams.

                                                                        $ \,$           $ \,$           $ \,$           $ 1 $           $ 1$ $ 0 $       $ 0$     $ 0 $             $ 2 $     $ 2$       $ 1\, $ $ 1$           $ 0\,$             $ 0 $         $ 0$ $ 0 $       $ 0$   $ 0 $     $ 0$     $ 0\, $ $ 0$         $ 0 $         $ 0$ $ 1 $     $ 1$     $ 0 $     $ 0$

The vacancy numbers have been written to the right of the parts (parts of the same size have the same vacancy). The shaded boxes are an example of a rigged configuration. Each such entry must not exceed the vacancy of that part. Furthermore, in diagram \(i,\) for sequence of block of parts of the same size \(j,\) the numbers in the shaded boxes are weakly decreasing, so that they form an integer partition \(I^i_j.\)

The pair \((\nuvec, I)\) is called a rigged configuration. The set of rigged configurations of type \((\lambda,\mu)\) are in bijection with \(\SSYT(\lambda,\mu).\) Given a configuration \(\nuvec,\) there are \(\prod_{i,j \geq 1} \binom{P_{ij} + Q_{ij}}{Q_{ij}}\) different rigged configurations.

#Charge

The charge of a configuration \(\nuvec\) is \[\charge(\nuvec) = \sum_{i,j} \binom{ m_{ij} }{2},\] and the charge of a rigged configuration is defined as \[\charge(\nuvec,I) = \charge(\nuvec) + \sum_{i,j} I^i_j.\]

Theorem (See [DLT94]).

The Kostka–Foulkes polynomial \(K_{\lambda\mu}(q)\) is equal to \[K_{\lambda\mu}(q) = \sum_{\nu} \prod_{i,j \geq 1} q^{\charge(\nu)} \qbinom{P_{ij} + Q_{ij}}{Q_{ij}}_q.\]

Further references include A. Schilling’s review of the \(X=M\) theorem and fermionic formulas [Sch07], her account of rigged configurations and the Bethe ansatz [Sch02], and work of A. N. Kirillov and R. Sakamoto relating rigged configurations to 10-eliminations [KS09].

Bibliography

  1. [DLT94]Jacques Désarménien, Bernard Leclerc and Jean-Yves Thibon. Hall-Littlewood functions and Kostka–Foulkes polynomials in representation theory. Séminaire Lotharingien de Combinatoire [electronic only], 32:38, 1994.
    .bib
    @article{DesarmenienLeclercThibon1994,
author = {Jacques D{\'{e}}sarm{\'{e}}nien and Bernard Leclerc and Jean-Yves Thibon},
journal = {S{\'{e}}minaire Lotharingien de Combinatoire [electronic only]},
language = {eng},
pages = {38},
publisher = {Universit{\"{a}}t Wien, Fakult{\"{a}}t f{\"{u}}r Mathematik},
title = {Hall-{L}ittlewood functions and {K}ostka--{F}oulkes polynomials in representation theory},
url = {http://eudml.org/doc/119019},
volume = {32},
year = {1994},
}
  2. [KR88]A. N. Kirillov and N. Yu. Reshetikhin. The Bethe ansatz and the combinatorics of Young tableaux. Journal of Mathematical Sciences, 41:925–955, 1988.
    .bib
    @article{KirillovReshetikhin1988thebethe,
  author = {A. N. Kirillov and N. Yu. Reshetikhin},
  title = {The {B}ethe Ansatz and the combinatorics of {Y}oung tableaux},
  journal = {Journal of Mathematical Sciences},
  year = {1988},
  volume = {41},
  pages = {925--955},
  issn = {1072-3374},
  issue = {2},
  keyword = {Mathematics and Statistics},
  publisher = {Springer New York},
  doi = {10.1007/BF01247088}
}
  3. [KS09]Anatol N. Kirillov and Reiho Sakamoto. Relationships between two approaches: Rigged configurations and 10-eliminations. Letters in Mathematical Physics, 89(1):51–65, 2009.
    .bib
    @article{KirillovSakamoto2009,
  author = {Kirillov, Anatol N. and Sakamoto, Reiho},
  title = {Relationships between two approaches: rigged configurations and
    10-eliminations},
  year = {2009},
  journal = {Letters in Mathematical Physics},
  volume = {89},
  number = {1},
  pages = {51--65},
  publisher = {Springer Science and Business Media LLC},
  doi = {10.1007/s11005-009-0318-3},
  url = {http://dx.doi.org/10.1007/s11005-009-0318-3},
  issn = {1573-0530}
}
  4. [Sch02]Anne Schilling. Rigged configurations and the Bethe ansatz. 2002.
    .bib
    @article{Schilling2002RiggedBethe,
  author = {Anne Schilling},
  title = {Rigged configurations and the {B}ethe ansatz},
  year = {2002},
  doi = {10.1142/9789812704474_0017},
  url = {https://doi.org/10.1142/9789812704474_0017}
}
  5. [Sch07]Anne Schilling. Part 4. ${X}={M}$ theorem: Fermionic formulas and rigged configurations under review. Combinatorial aspect of integrable systems:75–104, 2007.
    .bib
    @inbook{Schilling2007XM,
  author = {Schilling, Anne},
  title = {Part 4. ${X}={M}$ theorem: fermionic formulas and rigged
    configurations under review},
  year = {2007},
  pages = {75--104},
  publisher = {The Mathematical Society of Japan},
  doi = {10.2969/msjmemoirs/01701c040},
  url = {http://dx.doi.org/10.2969/msjmemoirs/01701c040},
  issn = {2189-1494},
  booktitle = {Combinatorial Aspect of Integrable Systems}
}

I use cookies to detect website issues and track search terms.