Tensor Power Asymptotics for Linearly Reductive Groups

Abstract: Given a finite-dimensional faithful representation $V$ of a linearly reductive group $G$ over a field $K=\bar K$, we consider the growth of the number of irreducible factors of $V^{\otimes n}$ when $n$ is large. We prove that there exist upper and lower bounds which are constant multiples of $n^{-u/2} (\dim V)^n$, where $u$ is the dimension of any maximal unipotent subgroup of $G$.

• The paper establishes explicit asymptotic bounds for the number of irreducible summands in tensor powers of a faithful representation of a linearly reductive group.
• It employs probabilistic methods, Edgeworth expansions, and structural reductions to connect representation growth with the dimension of maximal unipotent subgroups.
• The findings reveal optimal bounds up to constant factors, emphasizing intrinsic limitations on achieving exact leading-term asymptotics in various characteristic settings.

Asymptotic Growth of Irreducible Factors in Tensor Powers for Linearly Reductive Groups

Introduction

The paper "Tensor Power Asymptotics for Linearly Reductive Groups" (2512.22985) addresses the quantitative behavior of the number of irreducible summands in the tensor powers $V^{\otimes n}$ of a finite-dimensional faithful representation $V$ of a linearly reductive algebraic group $G$ defined over an algebraically closed field $K$. The author establishes upper and lower asymptotic bounds for the count of distinct irreducible summands as $n \to \infty$, providing a dimension-theoretic interpretation in terms of the group structure. This work closes a notable gap in the literature regarding the asymptotics of decompositions in tensor powers for general linearly reductive groups.

Main Results

The central theorem states that if $V$ is a faithful representation of a linearly reductive group $G$ and $u$ denotes the dimension of a maximal unipotent subgroup of $G$, there exist constants $A,B>0$ such that for all sufficiently large $n$,

$$A n^{-u/2} (\dim V)^n < b_n^{G,V} < B n^{-u/2} (\dim V)^n,$$

where $b_n^{G,V}$ denotes the cardinality of the set of irreducible $G$-summands in $V^{\otimes n}$.

This result is valid in slightly more general settings, specifically when $\ker \rho$ is central in $G$, with $\rho$ the representation. The asymptotic exponent $-u/2$ is shown to reflect the "deficit" arising from the group’s unipotent structure, aligning with expectations derived from the central limit and the Weyl dimension formulas.

The paper also asserts that no stronger asymptotic formula holds in general: for example, an explicit limit law of the form $b_n^{G,V} \sim C n^{-u/2} (\dim V)^n$ often fails, even in characteristic zero. The result reveals a sharp phase in the asymptotic regime, with the rate determined by the intrinsic algebraic group invariants.

Techniques and Proof Structure

The analysis proceeds through a sequence of structural reductions and analytic estimates:

1. Reduction to Connected, Reductive Case: Leveraging Clifford theory and the structure of linearly reductive groups, the argument proceeds under the assumption that $G$ is connected and reductive (or a torus in positive characteristic).
2. Weight Multiplicity Model via Probability: The decomposition of $V^{\otimes n}$ into weight spaces is modeled probabilistically, associating weights in the tensor power to sums of i.i.d. random variables, each distributed according to the formal character of $V$. The multiplicities correspond to probabilities of weight sums.
3. Asymptotic Expansion by Edgeworth and Local Limit Theorems: The local limit theorem and finite Edgeworth expansions provide precise asymptotic estimates for weight multiplicities. In the "central region" (weights of size $O(\sqrt{n})$), these estimates yield the dominant contribution.
4. Asymptotics of Irreducible Multiplicities: By repeatedly applying combinatorial difference operators (via the Weyl character formula and finite differences), the weight multiplicities are lifted to bounds for the irreducible character multiplicities.
5. Summation over the Dominant Chamber: The main term is estimated using polynomial bounds over cones in weight space, with positive-definite quadratic decay contributed by the group's root data.
6. Sharpness of Bounds: The results are shown to be optimal up to constant factors, tied directly to the geometry of the unipotent radical and the dimension data of $G$.

Numerical Strength and Contradictory Phenomena

A key numerical claim is the precise appearance of the exponent $-u/2$, where $u$ is the dimension of a maximal unipotent subgroup. This aligns with probabilistic interpretations of the distribution of weights and the geometry of representations, generalizing the $-1/2$ exponent seen in the classical $SL_2$ case.

Conversely, the paper disproves the existence of exact leading-term asymptotics in the general case (i.e., $b_n^{G,V} \sim C n^{-u/2} (\dim V)^n$ is not always correct), showing that fluctuations stemming from group structure and representation-theoretic data preclude such statements, even for well-understood finite groups and their natural representations.

Implications and Theoretical Significance

These results have several implications for the theory of algebraic and Lie group representations:

• Dimension-Theoretic Interpretation: The appearance of the unipotent dimension in the asymptotics provides a concrete connection between group structure and tensor power decomposition, facilitating explicit prediction of growth rates for semisimple group representations.
• Limiting Distribution Insight: The use of probabilistic and analytic approximation methods reflects a fruitful synthesis of combinatorial, geometric, and analytic perspectives on representation growth.
• Obstructions in Positive Characteristic: The impossibility of exact leading-term asymptotics in characteristic $p>0$ gives definitive constraints on potential extensions of classical limit laws in modular representation theory.

Practically, the explicit bounds for $b_n^{G,V}$ inform the asymptotic complexity of algorithms relying on tensor power decompositions (e.g., in computational invariant theory, quantum information theory, and categorical representation frameworks).

Prospects for Further Research

Future directions could include:

• Refinement for Non-Faithful or Non-Irreducible Cases: Extending the analysis to more general classes of representations, including tilting modules and indecomposable (but not irreducible) summands.
• Application to Tensor Categories: Investigating analogous asymptotics in wider tensor categories, with potential links to categorification and higher representation theory.
• Algorithmic and Computational Aspects: Leveraging these asymptotic bounds for improved design and analysis of algorithms in computational representation theory and related enumerative problems.

Conclusion

This work establishes sharp, explicit upper and lower bounds for the number of irreducible summands in high tensor powers of representations of linearly reductive groups, parametrized by the dimension of maximal unipotent subgroups. The combination of probabilistic weight analysis and group-theoretic structure theory provides a clear and general framework for understanding the asymptotic combinatorics of tensor powers, while delineating intrinsic limitations on the possibility of finer asymptotic formulas. The approach yields new foundational insights into the interface of algebraic group theory, representation theory, and asymptotic analysis.