Cores

# 1. Definition

A $k$-core is an integer partition $\lambda$ which has no hooks whose size is a multiples of $k$, or, equivalently, $\lambda$ has no rim hooks that are multiples of $k$ [JK81].

# 2. Properties

• For fixed $k \geq 1$, let $c_k(n)$ denote the number of $k$-cores of size $n$. It was shown in [Ono95] that the generating function for $c_k(n)$ is $$\sum_{n=0}^\infty c_k(n) q^n = \prod_{n=1}^\infty \frac{(1 - q^{kn})^k}{(1 - q^n)}.$$

• Since $2$-cores have triangle shaped Ferrers diagram, $c_2(n) = 1$ if $n= m(m+1)/2$ is a triangular number, and $c_2(n) = 0$ otherwise. This is,

$$\prod_{n=1}^\infty \frac{(1 - q^{2n})^2}{(1 - q^n)} = \sum_{n=0}^\infty c_2(n) q^n = 1 + q + q^3 + q^6 + q^{10} + q^{15} + q^{21} + \cdots$$

# 3. Statistics

We have the following 5 statistics on Cores in the database:

The length of a core.
The size of a core.
The number of strong covers of a core.
Number of covers of a core in weak Bruhat order.
The number of $k$-cores contained by a $k$-core.

# 4. Maps

We have the following 2 maps in the database:

to bounded partition
to partition
• There is a bijective correspondence between $k$-cores and $(k-1)$-bounded partitions.

The $(k-1)$-bounded partition corresponding to the $k$-core $\lambda$ is obtained by removing all the cells of $\lambda$ whose hook length is greater than or equal to $k$.

The $k$-core $\lambda$ is obtained from its corresponding $(k-1)$-bounded partition by finding the partition $\mu$ so that the skew partition $\lambda / \mu$ meets the following criteria:

1. Every row of $\lambda / \mu$ has the same number of cells as the $(k-1)$-bounded partition

2. No cell in $\lambda / \mu$ has a hook length exceeding $(k-1)$

3. Each empty cell above $\lambda / \mu$ has a hook length exceeding $(k-1)$. [LLM12]

• Example: $(5,4,2,1)$ is a $5$-core and has corresponding $4$-bounded partition $(3,3,2,1)$, which is found by removing all cells with hook length greater than 5 from $(5,4,2,1)$. Hence we remove 2 cells from row 1 and 1 cell from row 2.

•  $\begin{matrix} \bullet & \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet \\ \bullet \end{matrix} \quad \rightarrow \quad \begin{matrix} \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet \\ \bullet & \bullet \\ \bullet \end{matrix}$
• Example: $(3,3,2,1)$ is a $4$-bounded partition. The skew partition $(5,4,2,1)/(2,1)$ meets all the criteria above and we obtain $(5,4,2,1)$, the corresponding $5$-core.

•  $\begin{matrix} \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet \\ \bullet & \bullet \\ \bullet \end{matrix} \quad \rightarrow \quad \begin{matrix} & &\bullet&\bullet&\bullet\\&\bullet&\bullet&\bullet\\\bullet&\bullet\\\bullet \end{matrix} \quad \rightarrow \quad \begin{matrix} \bullet & \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet \\ \bullet \end{matrix}$
• There is a bijection between $k$-cores and affine Grassmannian elements. These can be identified by minimal length coset representatives of $\tilde{S}_k / S_k$.

For $k$-cores, we can map the corresponding $(k-1)$-bounded partitions to minimal length affine permutation representatives by using the content of the cells of the of the Ferrers diagram modulo $k$. This reduced content is called the residue of a cell. We read across the rows from right to left, starting at the bottom row and working up, and writing $s_i$ when the content of the cell mod $k$ is $i$.

For minimal length representatives of $\tilde{S}_k / S_k$, we apply the left action of $\omega \in \tilde{S}_k / S_k$ to the empty core to obtain a $k$-core. Define the left action by letting $s_i \cdot \lambda$ be the partition:

1. if there is at least one addable corner of residue $i$, then the result is $\lambda$ with all addable corners of $\lambda$ of residue $i$ added.

2. if there is at least one removable corner of residue $i$, then the result is $\lambda$ with all removable corners of $\lambda$ of residue $i$ removed.

3. otherwise the result is $\lambda$. [LLM12]

• Example: Take the $5$-core $(5,4,2,1)$. It maps to the $4$-bounded partition $(3,3,2,1)$. We label the content of the cells mod $5$ and write down the corresponding permutation.

•  $\begin{matrix}0&1&2\\4&0&1\\3&4\\2\end{matrix} \to s_2s_4s_3s_1s_0s_4s_2s_1s_0$
• Example: Take the affine permutation representative $s_2s_4s_3s_1s_0s_4s_2s_1s_0$ and apply the left action to the empty core to obtain the $k$-core $(5,4,2,1)$.

$$s_2s_4s_3s_1s_0s_4s_2s_1s_0 \cdot \emptyset \to s_2s_4s_3s_1s_0s_4s_2s_1 \cdot \left\lvert\begin{matrix}\hline 0\\ \hline\end{matrix}\right\rvert \\ \to s_2s_4s_3s_1s_0s_4s_2 \cdot \left\lvert \begin{matrix}\hline 0&1\\ \hline\end{matrix}\right\rvert \to s_2s_4s_3s_1s_0s_4 \cdot \left\lvert \begin{matrix}\hline 0&1&2\\ \hline\end{matrix}\right\rvert \\ \to s_2s_4s_3s_1s_0 \cdot \left\lvert \begin{matrix}\hline 0&1&2\\4\\ \hline\end{matrix}\right\rvert \to \cdots \to \left\lvert \begin{matrix}\hline 0&1&2&3&4\\4&0&1&2\\3&4\\2\\ \hline\end{matrix}\right\rvert$$

# 5. Covers and Ordering

Note we define a $k$-core's length as the number of elements with a hook length of less than $k$, or equivalently the size of the corresponding $(k-1)$-bounded partition or the length of the corresponding affine permutation.

## 5.1. Weak Covers

We say that the $k$-core $\lambda$ covers the $k$-core $\mu$ if, and only if, the length of $\lambda$ is one more than the length of $\mu$ and there exists some $i$ such that $\lambda=s_i \cdot \mu$. This defines a weak Bruhat order on $k$-cores. [LLM12]

• Example: The $5$-core $(6,4,3,1)$ covers the $5$-core $(5,4,2,1)$. The length of $(5,4,2,1)$ is 9, and the length of $(6,4,3,1)$ is 10.

• $s_0\cdot\left\lvert\begin{matrix}\hline 0&1&2&3&4\\4&0&1&2\\3&4\\2\\\hline\end{matrix}\right\rvert\to\left\lvert\begin{matrix}\hline 0&1&2&3&4&0\\4&0&1&2\\3&4&0\\2\\\hline\end{matrix}\right\rvert$

## 5.2. Strong Covers

We say that the $k$-core $\lambda$ covers the $k$-core $\mu$ if, and only if, the length of $\lambda$ is one more than the length of $\mu$ and $\mu$ is contained in $\lambda$. This defines a strong order on $k$-cores. [LLM12]

• Example: The $5$-core $(8,4,2,1)$ covers the $5$-core $(5,4,2,1)$.

•  $\begin{matrix} \bullet & \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet \\ \bullet \end{matrix}$ $\quad \subset \quad$ $\begin{matrix} \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet \\ \bullet \end{matrix}$

# 6. $(a,b)$-cores

## 6.1. Definitions

• $\lambda$ is an $(a,b)$-core $\lambda$ if, and only if, it is both an $a$-core and a $b$-core.

• Consider the boxes in the first column of $\lambda$ and reduce their hook lengths modulo $a$. Consider the highest row in each residue class. These are the $a$-rows of the diagram.

• Example: $\lambda = (5,2,1,1,1)$ is a (4,7)-core, hence the 4-rows are of $\lambda$ are rows 1, 3, and 4.

•  $\begin{matrix} 1 & \bullet & \bullet & \bullet & \bullet & \bullet \\ 1 & \bullet & \bullet \\ 3 & \bullet \\ 2 & \bullet \\ 1 & \bullet \end{matrix}$
• The $b$-boundary of an $(a,b)$-core $\lambda$ is all the cells with hook length less than $b$.

• Example: $\lambda = (5,2,1,1,1)$ is a (4,7)-core, it has $b$-boundary

•  $\begin{matrix} & \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet \\ \bullet \\ \bullet \\ \bullet \end{matrix}$
• The skew length of an $(a,b)$-core is the number of cells in the $a$-rows and the $b$-boundary. Hence $\lambda$ from the previous examples has skew length 6 since there are 6 cells in the $1^{st}$, $3^{rd}$, and $4^{th}$ row of the $b$-boudnary.

## 6.2. Properties

• There multiple open questions and conjectures concerning simultaneous $(a,b)$-cores relating to generalized Catalan numbers and $q,t$-Catalan numbers. See Results and conjectures on simultaneous core partitions at arXiv:1308.0572. [AHJ14]

• The total number of $(a,b)$-core partitions is finite if, and only if, $a$ and $b$ are coprime, in which case the number is $$\frac{1}{a+b} \binom{a+b}{a,b} = \frac{(a+b-1)!}{a!b!}$$

[And02]

# 7. References

[AHJ14]   Armstrong, D., Hanusa, C. R. H., Jones, B. C., Results and conjectures on simultaneous core partitions, arXiv:1308.0572 (2014).

[And02]   Anderson, Jaclyn, Partitions which are simultaneously $t_1$- and $t_2$-core, Disc. Math., 248, (2002), pp. 237-243.

[JK81]   James, Gordon; Kerber, Adalbert, The representation theory of the symmetric group, Addison-Wesley Publishing Co., Reading, Mass. (1981).

[LLM12]   Lam, T., Lapointe, L., Morse, J., et al., k-Schur functions and affine Schubert calculus, arXiv:1301.3569 (2012).

[Ono95]   Ono, Ken, A Note on the number of $t$-Core Partitions, Rocky Mountain Journal of Mathematics 25 (1995), no. 3, pp. 1165-1169.