# The symmetric functions catalog

An overview of symmetric functions and related topics

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.

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 $K_{\lambda\mu}(q) = \sum_{\nu} \prod_{i,j \geq 1} q^{\charge(\nu)} \qbinom{P_{ij} + Q_{ij}}{Q_{ij}}_q.$