2019-07-01

## Rigged configurations

Rigged configurations were introduced by Kirillov and Reshetikhin [KR88].
One of the main applications is to compute Kostka–Foulkes polynomials in an efficient manner.
We shall now introduce the terminology to define these objects, by following the
introduction in [DLT94].
Let $\lambda$ and $\mu$ be partitions of $n.$ A *matrix of type* $(\lambda,\mu)$
is a matrix $M=(m_{ij})$ with a finite number of non-zero in $\setZ,$ 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 $M,$ via \[ P_{ij} = \sum_{k \leq j } \left( m_{i,k} - m_{i+1,k} \right) \qquad Q_{ij} = \sum_{k \geq i+1 } \left( m_{k,j} - m_{k,j+1} \right). \]

The matrix $M$ is *admissible* if and only if the entries in $P$ and $Q$ are non-negative.
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 $i,$ $j,$ an *integer partition* $I^i_j$
is assigned to each $j$ which appear as a part of $\nu^i,$
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. \[ M= \begin{pmatrix} 3 & 2 & 1 & -1 & 1 & 1 \\ 2 & 1 & 1 & 1 & 0 & 1 \\ 2 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \end{pmatrix} \]

\[ P= \begin{pmatrix} \mathbf{1} & 2 & \mathbf{2} & \mathbf{0} & 1 & \mathbf{1} \\ \mathbf{0} & 0 & \mathbf{0} & \mathbf{0} & \mathbf{0} & 1 \\ 1 & 1 & \mathbf{1} & 1 & \mathbf{0} & 0 \\ 0 & 0 & \mathbf{0} & 1 & 2 & 2 \\ 1 & 2 & 3 & 3 & 3 & 3 \\ \end{pmatrix} \; Q= \begin{pmatrix} 2 & 0 & 1 & 2 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{pmatrix} \]We have that 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 is 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 \[ K_{\lambda\mu}(q) = \sum_{\nu} \prod_{i,j \geq 1} q^{\charge(\nu)} \qbinom{P_{ij} + Q_{ij}}{Q_{ij}}_q. \]

## References

- [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.
- [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.