Dimension statistics of representations of finite groups

TL;DR

This paper analyzes how the dimension data of irreducible representations of finite groups relate to conjugacy-class and orbit data across several group families. Using Kirillov theory for nilpotent groups, unipotent wavefront data for reductive groups, and probabilistic/thermodynamic limits for symmetric groups, it demonstrates concentration phenomena for reductive groups over large finite fields and for GL_n(F_q) as n grows, while showing that S_n under the uniform measure exhibits a spread of dimensions not seen in the reductive cases. The results reveal a nuanced landscape: dimensions often cluster near maximal scales in some families, yet in S_n the spectrum is broad and only loosely correlated with class sizes. The work also introduces asymptotic frameworks (collinearity and related notions) to compare dimensional data with orbit- or class-size data, and highlights open questions about when and why such correspondences hold. Overall, the paper advances a geometric-statistical perspective on representation-theoretic dimension data across finite groups and points to deeper structural links between representations and orbit structures.

Abstract

This paper discusses what the dimension data of irreducible representations of a finite group looks like in some specific cases, including unipotent and reductive groups over finite fields. The essence of our investigation is whether the dimension data of irreducible representations of a finite group can be ``geometerized'', to become equal to the cardinality of certain orbit spaces. The first part of this paper deals with nilpotent and reductive groups over finite fields, whereas the second part deals with the symmetric group $S_n$. The main conclusion that we want to bring out to contrast these two cases is that for reductive groups over finite fields, the dimension data is concentrated (in a statistical sense) in a neighborhood of the maximal dimension, whereas for the symmetric group, it is spread out.

Figures (5)

• Figure 1: Histogram of dimensions of irreducible representations of ${\rm S}_{20}$ and ${\rm S}_{30}$. The $x$-axis on the left (resp. right) plot is scaled by $10^7$ (resp. $10^{13}$).
• Figure 2: A comparative logarithmic plot of the average dimension of a uniformly random partition of size $n$ (in blue), the maximum dimension of a partition of $n$ (in magenta) as $n$ varies from $1$ to $30$, and the asymptotic average dimension (in red).
• Figure 3: Comparative histogram of log of squares of dimensions and class sizes of ${\rm S}_{20}$ and ${\rm S}_{30}$. The yellow part is $\ln c_\lambda$, the blue part is $\ln (d_\lambda^2)$ and the intersection is coloured green. The number of bins has been specified to be $20$ and $30$ respectively.
• Figure 4: The ratio $|A^{n} [\alpha, \beta]|/|B^{n} [\alpha, \beta]|$ plotted for $\alpha = 0.4$ and $\beta = 0.8$ on the left, and $\alpha = 0.2$ and $\beta = 0.6$ on the right, for $n$ ranging from $5$ to $40$.
• Figure 5: A joint list plot of $a^{n,k}$ (in orange) and $b^{n,k}$ (in blue) for $k$ ranging from $1$ to $n$, with $n = 20$ on the left and $n = 40$ on the right.

Theorems & Definitions (27)

• Theorem 2.1: Kirillov
• Definition 3.1: Selfdual nilpotent group
• Remark 3.2
• Theorem 4.1
• Theorem 4.2
• Definition 5.1
• Definition 5.2
• Definition 5.3
• Remark 5.4
• Theorem 6.1