Tensor Powers of Representations

• Tensor powers of representations are iterated tensor products that capture structural and categorical properties of modules.
• They provide insight into decomposition patterns and asymptotic growth, exemplified by exponential behavior in indecomposable summands.
• Applications span group, quantum, and diagram algebra representations, utilizing combinatorial and spectral techniques for precise calculations.

A tensor power of a representation refers to the iteration of the tensor product operation on a fixed module or representation within a suitable tensor category, often with the aim of understanding structural, asymptotic, or categorical properties of representations and their associated algebras. Across a broad range of algebraic settings—including group representations, Hopf algebras, quantum groups, semigroups, and monoidal diagram algebras—studying tensor powers provides insight into decomposition patterns, spectral asymptotics, and equivalence or duality phenomena. The following sections survey central principles, characterizations, computational results, and applications as found in the technical literature.

1. Structural Definitions and Categorical Formulations

A tensor power $V^{\otimes n}$ of a module (or general representation) $V$ refers to the $n$-fold iterated tensor product in the relevant monoidal category. In the representation category of a Hopf algebra $H$, this is the iterated $H$-module

$$V^{\otimes n} = V \otimes \cdots \otimes V \quad (n \text{ times}),$$

with the diagonal action $$h \cdot (v_1 \otimes \cdots \otimes v_n) = \sum h_{(1)} \cdot v_1 \otimes \cdots \otimes h_{(n)} \cdot v_n$$ (with Sweedler notation). In settings such as group representations or semigroup modules, the same construction encodes composition laws and branching information.

The paper of tensor powers is tightly connected to the notion of depth (in the theory of subalgebras), spectral characteristics (as in growth, covering, and mixing phenomena), centralizer algebra dualities (Schur–Weyl type theorems), and asymptotic decomposition patterns (for example, Plancherel growth and statistical measures on component multiplicities).

In settings where the structure is not semisimple, indecomposable direct sum decompositions replace sums of simples, and the Krull–Schmidt property becomes vital; this is especially so in categories such as finite-dimensional modules for (super)groups, quantum groups at roots of unity, finite monoids, and diagram algebras (Coulembier et al., 2023, He et al., 6 Aug 2025).

2. Decomposition Patterns and Asymptotics

A recurrent theme in modern research is the description of direct sum composition of tensor powers. This is captured by statistics such as:

• $l(n)$: total length (sum of composition factors, with multiplicity) of $V^{\otimes n}$.
• $b(n)$: total number of indecomposable summands (with multiplicity) in $V^{\otimes n}$.

For a module $V$ of dimension $d$, it is established under wide generality (including finite, algebraic, and monoidal supergroup schemes, quantum groups, semigroups, and Tannakian categories) that the growth rate

$\beta = \lim_{n \to \infty} (b(n))^{1/n}$

always satisfies $\beta = d$ (Coulembier et al., 2023). The exponential growth rate of the number of indecomposable summands thus attains the trivial upper bound given by the dimension of the underlying space, and this behavior remains robust even in wild and non-semisimple settings.

For finite monoids and diagram algebras (such as Temperley–Lieb, Motzkin, and planar rook monoids), explicit formulas tie the length and number of indecomposables to structural eigenvalues of representation matrices; in planar rook, for instance, the character table format is a truncated Pascal triangle (He et al., 6 Aug 2025). Cellular and diagrammatic approaches (for example, in TL, Brauer, and Khovanov algebras) allow the explicit calculation of multiplicities and yield spectral gap estimates governing the subleading behavior.

In group representations, similar exponential growth is seen in the number of direct sum components, with explicit polynomial correction terms visible in the $SL_2$ case: the number of indecomposables grows as $2^n / \sqrt{n}$, matching the binomial coefficient structure in the character expansion (Coulembier et al., 2023).

3. Combinatorial and Spectral Formulas

The decomposition of tensor powers frequently reduces to combinatorial expressions involving binomial coefficients, Stirling numbers, Kostka numbers, and related invariants tied to the structure theory of the algebra in question. In semisimple settings such as symmetric groups, irreducible multiplicities in tensor powers of the defining module are given explicitly via

$$a_{\lambda,n} = \sum_{i=|\overline{\lambda}|}^n \binom{i}{|\overline{\lambda}|} \left\{ n \atop i \right\} f^{\overline{\lambda}},$$

where $\left\{ n \atop i \right\}$ denotes Stirling numbers of the second kind and $f^{\overline{\lambda}}$ is the number of standard Young tableaux of the subpartition, linking to set partitions and enumeration (Ding, 2015, Ding, 2013).

For finite monoids with a group of units $G$, spectral techniques utilize character tables of the $G$-modules and (cell) decomposition matrices to express asymptotic lengths: $$l(n) \sim \sum_{g \text{ scalar}} \frac{|C_g| T_g (\chi(g))^n}{|G_g|},$$ where $C_g$ denotes $p$-regular conjugacy classes, $T_g$ is given by the inverse decomposition matrix, and $\chi(g)$ is the eigenvalue of $g$ on $V$ (He et al., 6 Aug 2025).

In the case of diagram monoids and their associated cellular algebras, further combinatorial refinements exploit triangle-like or Riordan array inversion for multiplicities; for the planar rook monoid, with simple characters $\chi_i(j) = \binom{j}{i}$, the multiplicities in tensor powers correspond exactly to such combinatorial triangles.

4. Tensor Powers in Group and Monoid Representation Theory

Tensor powers provide a framework for analyzing asymptotic and probabilistic properties:

• In symmetric group theory, near-universal "covering" properties are established: with high probability, the tensor products of randomly chosen irreducibles will contain all irreducible components, and minimal tensor power required to achieve full support is governed by $\sim (n\log n)/\log(\dim V)$ (Sellke, 2020).
• For monoids, analogues of Pontryagin's results for groups show that, eventually, almost all direct summands of tensor powers are induced from the group of units $G$, and in certain diagram algebras, almost all indecomposable components stabilize to a predicted structure (He et al., 6 Aug 2025).
• Growth in the sum of squares of degrees of irreducible constituents under tensor powers, as measured by the Plancherel norm, exhibits "2-step" growth, with explicit power growth lower bounds, in finite simple groups of Lie type and compact semisimple Lie groups (Larsen et al., 2021).

5. Explicit Computations for Diagram Monoids

The paper (He et al., 6 Aug 2025) gives detailed computations for several diagram monoids:

Monoids	Simple Characters	Dominant Eigenvalue	Multiplicity Formula Shape
Planar rook (pRo$_m$)	$\chi_i(j) = \binom{j}{i}$	$d$ (dim $V$)	Pascal triangle, binomial inversion
Temperley–Lieb (TL$_m$)	$\alpha_{j,i}$ formula	$d$	Catalan-like, generating function inversion
Motzkin (Mo$_m$)	$\beta_{j,i}$ formula	$d$	Motzkin triangle, combinatorial summations

These algebraic and combinatorial structures allow the calculation of length and number of summands in $V^{\otimes n}$ and provide a classification of asymptotic behavior. Eigenvalues of the fusion (or action) matrix play a crucial role; the largest eigenvalue dictates the leading exponential growth, while the decomposition matrix structure details the subleading corrections.

6. Applications, Implications, and Conjectures

The universal asymptotic result $\beta = \dim V$ highlights a fundamental law: in orderly or wild settings, with or without semisimplicity, the "size" (number of summands) of the $n$-fold tensor power of a $d$-dimensional representation grows like $d^n$. This has multiple ramifications:

• Provides tools for efficient computational representation theory (since explicit formulas and spectral techniques enable practical calculation of complex decompositions for large $n$).
• Supports conjectures on induction-restriction and structural stability in the context of group and semigroup representations (He et al., 6 Aug 2025).
• Establishes, in diagrammatic categories, deep combinatorial correspondences—with connections to classical combinatorics (Catalan, Motzkin, Pascal) and to categorification.

Open problems and plausible extensions involve finer classification in wild representation type, comparison between monoidal and supermonoidal categories, and explicit descriptions of the "group-injective" direct summand phenomenon (i.e., determining the threshold $n$ such that $V^{\otimes n}$ contains a summand induced from the group of units) (He et al., 6 Aug 2025).

7. Further Directions

The recent advances surveyed here extend the analysis of tensor powers:

• Beyond semisimple categories, providing growth and decomposition invariants in quantum, super, and diagrammatic monoid contexts (Coulembier et al., 2023, He et al., 6 Aug 2025).
• Via explicit combinatorial models, as in the use of lattice-path counting for quantum group tensor powers (Lachowska et al., 5 Apr 2024).
• Through probabilistic and asymptotic methods, which open the path to applications in complexity theory, random walks on groups, and statistical representation theory (Sellke, 2020, Larsen et al., 2021).

Thus, the complex algebraic machinery underlying tensor powers admits comprehensive analysis through categorical, combinatorial, and spectral lenses, yielding both explicit enumeration and deeper understanding of structural invariants in representation theory.