SL₂ tensor spectrum

The spectrum of tensor-power growth for the natural module of SL₂(2ʳ)

Pranava Kumar, Taksh Kothari

Research manuscript  ·  MSC 2020: 20C20, 20G05, 20G40

The complete spectrum

Move through the family. Every mark is an eigenvalue of the action matrix.

q = 8 · 15 vertices
projective transient
Projective and transient eigenvalues A number line from minus two to two. Projective eigenvalues appear above the axis and transient eigenvalues below it.
projective eigenvalue 2 the unique eigenvalue of modulus 2
leading scale
2
first correction
1.8794
transient edge
1.8478
spectral gap
0.0316
The projective cell supplies the first correction, while the nonprojective part supplies the next spectral scale. Select any mark with a mouse or keyboard to read its exact formula.

Where the question starts

Take the two-dimensional natural module $V$ of the finite group $\SL_2(q)$ over a field of characteristic $2$, with $q=2^r$. Tensoring $V$ with itself produces larger modules, and those modules split into indecomposable direct summands. We write $b(n)$ for the number of summands in $V^{\otimes n}$, counted with multiplicity.

Characters record composition factors, which need not determine this direct-sum decomposition in modular representation theory. The fusion graph retains the missing information: its vertices are the indecomposable summands that occur, and a directed edge records what appears after one more tensor product with $V$. If $M$ is its action matrix, then

\[ b(n)=\mathbf 1^{\mathsf T}M^n e_0, \]

where $e_0$ starts at the trivial module. The growth problem therefore becomes a spectral problem, provided one can first determine the graph and its vertices.

indecomposable
a module that cannot be written as a direct sum of two nonzero modules
fusion graph
the transition graph for tensoring every indecomposable by $V$
projective cell
the closed part of the graph formed by projective indecomposable modules
transient spectrum
eigenvalues from the nonprojective blocks, which eventually feed into the projective cell

What is determined

For every $r\geq 1$, the fusion graph is finite and its full action matrix can be written explicitly. The description separates the nonprojective modules from the projective ones without discarding the arrows between them.

The fusion graph

The graph has exactly $2q-1$ vertices:

\[ T(0)|_{\SL_2(q)},\ldots,T(q-2)|_{\SL_2(q)} \qquad\text{and}\qquad P(S),\quad S\subseteq\mathbf Z/r\mathbf Z. \]

The first $q-1$ vertices are indecomposable and nonprojective; the remaining $q$ are the projective indecomposable modules.

The spectrum

The $2q-1$ eigenvalues are simple. The projective family consists of the Brauer character values

\[ 2,\qquad 2\cos\frac{2\pi l}{q-1},\qquad 2\cos\frac{2\pi m}{q+1}, \]

over the ranges in Theorem 1.2, while the transient family is $\{2\cos(j\pi/q):1\leq j\leq q-1\}$. The live number line above draws these formulas directly.

The first correction

For $r\geq2$, the count has the asymptotic form

\[ b(n)=\frac{3^r}{q(q^2-1)}2^n+\frac{S_q}{q+1}\left(-2\cos\frac{\pi}{q+1}\right)^n+O_q\!\left(\left(2\cos\frac{\pi}{q}\right)^n\right), \]

where $S_q$ is a negative algebraic integer of degree $\varphi(q+1)/(2r)$. The negative eigenvalue makes the correction alternate: for all sufficiently large $n$, the exact count lies above the leading term at odd $n$ and below it at even $n$.

A tensor product becomes a cyclic carry

Donkin's tensor product theorem labels the relevant tilting states by digits in $\{0,1,2\}$. Tensoring by $V$ increments the first digit; a $2$ branches and carries to the next position, and the last position carries back to the first.

Cyclic digit state Digits are arranged around a circle. The active path shows how tensoring by V carries through positions.

current state (0, 0, 0, 0)

The first digit is below 2, so it increments without a carry.

A branch with multiplicity $2$ remains at the current digit. The other branch carries clockwise. At most one complete turn is needed.

The page uses the same recurrence as code/tensor.py. The wheel exposes the cyclic step that is easy to miss in a linear digit string.

The fifteen-vertex graph for SL₂(8)

At $r=3$, the general construction recovers the fifteen-vertex example that motivated the calculation. The upper row contains $T(0),\ldots,T(6)$; the lower row contains the eight projective indecomposables. Projective vertices never point back to the transient row.

n = 0 · b(n) = 1
Fusion graph for the natural module of SL2(8) Seven nonprojective vertices appear above eight projective vertices. Directed edges show tensoring by the natural module. Node size changes with tensor-power multiplicity.
Node area records the current multiplicity in $V^{\otimes n}$; arrows and labels record the fixed action matrix. The animation changes the distribution, not the graph.

How the count approaches its leading term

The exact integer count comes from the cyclic state rule. The chart compares it with $a(n)=3^r2^n/[q(q^2-1)]$ and keeps the sign of the error visible.

Signed error in the tensor-power summand count A line plot shows the signed relative error between the exact count and its leading asymptotic term for tensor powers one through thirty-six.
tensor power
18
exact b(n)
—
leading a(n)
—
relative error
—
Blue marks odd powers and graphite marks even powers. In nth-root mode, the absolute height tends toward $\cos(\pi/(q+1))$; the sign still records which side of the leading term contains the exact count.

The route through the proof

  1. Encode tensoring.The identity $V^{\otimes3}\cong(V\otimes V^{[1]})\oplus V^{\oplus2}$ turns tensoring into the carry rule above.
  2. Identify the vertices.A weight bound preserves indecomposability for states with a zero digit. Full-length states are handled by factoring off the Steinberg module and comparing dimensions.
  3. Diagonalize both parts.A quadratic recursion yields Chebyshev polynomials for the transient blocks, while Brauer character duality supplies the projective spectrum and its coefficients.
  4. Compare the scales.The projective value $-2\cos(\pi/(q+1))$ lies just beyond the transient edge $2\cos(\pi/q)$ and controls the first correction.

Scope and computational evidence

The uniform results concern the natural module of $\SL_2(2^r)$ in characteristic $2$. The paper discusses the odd-characteristic tensor rule but does not carry out the spectral comparison there.

The supplementary program checks reachability, projective multiplicities, the regular-module dimension identity, and minimal polynomials through $r=8$. Through $r=6$, it also checks matrix dimensions, nonnegative entries, characteristic polynomials, and summand counts through $n=60$. These calculations reproduce examples and identities; the arguments establishing the uniform results remain in the manuscript.

PathWhat it contains
paper/the complete LaTeX source
code/tensor.pycarry rule, decompositions, action matrices, and exact counts
code/spectra.pycharacteristic and minimal polynomials, coefficients, and spectral terms
code/check.pythe reproducibility checks summarized above

Citation

@misc{kumar2026tensor,
  title  = {The spectrum of tensor-power growth for the natural module of $SL_2(2^r)$},
  author = {Kumar, Pranava and Kothari, Taksh},
  year   = {2026}
}