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Computing same-degree representations
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‎content/blog/2026-06-04-representations-with-the-same-degree.md‎

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---
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title: "Representations with the same degree"
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slug: "representations-with-the-same-degree"
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date: 2026-06-04
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author: "Pieter Belmans"
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---
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Time for the next installment of the arXiv series.
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Today's preprint is from earlier this year, and it's
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[Frank Lübeck: Representations with the same degree](https://arxiv.org/abs/2601.18786).
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In fact, whilst preparing this blogpost, I discovered that
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this question was also already addressed by Andy Huchala
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in [his 2018 undergraduate thesis](https://ahuchala.com/files/undergrad/Lie_Algebra_Representation_Thesis.pdf).
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In what follows, I will (mostly) refer to the arXiv preprint.
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As before, we will use [Semisimple.jl](https://homogeneous.tools/Semisimple.jl/dev/)
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to play with the constructions in the paper.
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## Infinitely many pairs
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**Theorem**
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_Let $G$ be a connected reductive simply-connected algebraic group over $\mathbb{C}$ of rank $\geq2$.
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Then $G$ has infinitely many pairs of irreducible rational representations $\rho_1,\rho_2$
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that are not related by an automorphism of $G$
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but for which $\dim\rho_1=\dim\rho_2$._
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The rank-$1$ assumption is necessary:
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in type $\mathrm{A}_1$ the irreducible representations are determined by their dimension.
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Beyond that, Lübeck establishes the theorem by exhibiting one explicit same-degree pair in every simple type,
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and combining this with the following observation:
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the Weyl dimension formula immediately gives
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$$\dim\mathrm{V}(k(\lambda+\rho)-\rho)=\dim\mathrm{V}(\lambda)\cdot k^N$$
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for any $k\in\mathbb{Z}_{>0}$,
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where $N$ is the number of positive roots.
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So one same-degree pair $(\lambda,\mu)$ produces infinitely many at degrees $d, d\cdot 2^N, d\cdot 3^N,\ldots$
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This is a **perfect** paper to play around with in Semisimple.jl,
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because the only ingredient we really need is `degree`.
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## Proposition 2: an explicit pair in each exceptional type, and in A₂, B₂
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Lübeck's Proposition 2 collects the following same-degree pairs,
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found by brute force over the dominant weights with small coordinates:
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```julia
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using Semisimple
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# (type, λ, μ, common degree)
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explicit = [
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(TypeA{2}, [1, 2], [0, 4], 15),
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(TypeB{2}, [1, 2], [0, 4], 35),
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(TypeG2, [3, 0], [0, 2], 77),
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(TypeF4, [1, 0, 0, 1], [2, 0, 0, 0], 1053),
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(TypeE{6}, [2, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], 351),
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(TypeE{7}, [0, 0, 0, 1, 1, 0, 0], [0, 0, 0, 0, 0, 2, 3], 1903725824),
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(TypeE{8}, [1, 0, 1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 1, 1], 8634368000),
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]
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for (DT, λ, μ, d) in explicit
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@assert degree(DT, λ) == degree(DT, μ) == d
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end
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```
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Note that in type $\mathrm{E}_6$ the non-trivial diagram automorphism
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gives a _second_ same-degree pair at $d=351$,
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namely $\mathrm{V}(2\omega_6)$ and $\mathrm{V}(\omega_5)$;
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the other six pairs have no such automorphic explanation.
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One can reproduce this brute-force search using Semisimple.jl:
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```julia
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# diagram automorphism on coordinates; trivial except for A_2 and E_6.
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outer(::Type{TypeA{2}}, v) = reverse(v)
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outer(::Type{TypeE{6}}, v) = (v[6], v[2], v[5], v[4], v[3], v[1])
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outer(::Type{<:DynkinType}, v) = v
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# smallest dimension realised by two dominant weights of DT
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# in different outer-automorphism orbits, searching coordinates in 0:N.
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function smallest_pair(::Type{DT}; N=4) where {DT<:DynkinType}
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l = rank(DT)
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bins = Dict{BigInt,Vector{NTuple{l,Int}}}()
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for v in Iterators.product(ntuple(_ -> 0:N, l)...)
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push!(get!(() -> NTuple{l,Int}[], bins, degree(DT, collect(v))), v)
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end
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pairs = [(d, ws) for (d, ws) in bins
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if length(unique(min(v, outer(DT, v)) for v in ws)) >= 2]
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return argmin(first, pairs)
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end
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# the cases considered in Proposition 2
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for DT in (TypeA{2}, TypeB{2}, TypeG2, TypeF4, TypeE{6}, TypeE{7}, TypeE{8})
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d, ws = smallest_pair(DT)
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println(rpad(string(DT), 12), " d = ", lpad(string(d), 12), " ", join(ws, " "))
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end
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```
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## Theorem 3: infinite families in the classical types
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For the classical types Lübeck exhibits closed-form families.
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In types $\mathrm{A}_l,\mathrm{B}_l,\mathrm{D}_l$ the pair is always of the shape
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$\mathrm{V}\bigl((c-1)\omega_2\bigr)$ vs.&nbsp;$\mathrm{V}\bigl(\omega_1+(c-2)\omega_2\bigr)$,
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where $c$ depends on the type;
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verifying them up to any reasonable rank is straightforward.
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For $\mathrm{A}_l$ it would be as in Theorem&nbsp;3(a):
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```julia
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for l in 2:15
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λ = zeros(Int, l); λ[2] = l - 1
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μ = zeros(Int, l); μ[1] = 1; μ[2] = l - 2
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expected =
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BigInt(2l - 1) *
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prod(BigInt(k)^2 for k in (l + 1):(2l - 2); init=BigInt(1)) ÷
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factorial(BigInt(l - 1))^2
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@assert degree(TypeA{l}, λ) == degree(TypeA{l}, μ) == expected
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end
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```
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The same pattern works for $\mathrm{B}_l$ and $\mathrm{D}_l$ with the closed forms
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in Theorem&nbsp;3(b) and 3(c) of the paper.
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Type $\mathrm{C}_l$ takes an unexpectedly arithmetic turn:
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Lübeck shows that, again for $\lambda=a\omega_1+b\omega_2$ and $\mu=(a-2)\omega_1+(b+1)\omega_2$,
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the equality $\dim\mathrm{V}(\lambda)=\dim\mathrm{V}(\mu)$ holds
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if and only if the generalised Pell equation
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$$c^2-(4l-5)\,a^2=(2l-3)^2$$
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admits a solution with $a\geq 3$,
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in which case $b=\tfrac12(c+1-a-2l)$.
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Because $4l-5\equiv 3\pmod 4$ is never a perfect square,
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this Pell equation has infinitely many solutions for every $l\geq 3$,
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and the construction has the curious feature that the smallest $(a,b)$ one obtains
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can be _enormous_:
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for $l=159$ the corresponding common degree has 15728 decimal digits.
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Computing that degree directly in Semisimple.jl just works,
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since `degree` returns `BigInt`.
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## Some comments
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* The seven Proposition 2 pairs, together with the three Theorem 3 families,
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are part of the Semisimple.jl test suite,
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giving a slightly unusual but pretty cool set of unit tests!
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* Huchala claims in his Theorem 7 that there is an upper bound on how many irreducible representations of a given dimension can exist in type $\mathrm{B}_2$ and $\mathrm{G}_2$. For $\mathrm{A}_2$, a fun argument using the rank of an elliptic curve (explicit in Huchala's thesis, whilst attributed to Deligne but not given in Lübeck's preprint) shows that for any positive integer $m$, you can find at least $m$ non-isomorphic irreducible representations of the same degree.
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However, the proof of the upper bound in types B and G is not correct: there is a plane curve _for every given degree_, on which Faltings' theorem gives a finite number of points.
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But to prove the claim, one would need that there is a uniform bound on the number of points all those plane curves, simultaneously.
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So there seems to be a fun open number-theoretical question here,
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unless I'm missing something?
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* There are fun OEIS sequences to add here:
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[A000891](https://oeis.org/A000891) are the smallest degrees in type $\mathrm{A}_l$ where duplicate representations exist,
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but the sequences in types B and D are missing (in type C the chaotic behavior makes it hard to produce the sequence).
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The entries for these sequences are computed both by Lübeck and Huchala.
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If anyone is interested in adding these, here is some Julia code to determine them:
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```julia
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function dim_B(l::Integer)
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@assert l >= 2
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l == 2 && return 35
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l = BigInt(l)
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num = 3 * (4l - 5) * (6l - 5) * (6l - 7) *
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prod(k^2 for k in (2l):(4l - 6); init=BigInt(1))
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den = factorial(2l - 3)^2
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q, r = divrem(num, den); @assert iszero(r)
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return q
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end
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function dim_D(l::Integer)
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@assert l >= 4
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l = BigInt(l)
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num = 3 * (3l - 4) * (3l - 5) * (4l - 7) *
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prod(k^2 for k in (2l - 1):(4l - 8); init=BigInt(1))
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den = (l - 2)^2 * factorial(2l - 5)^2
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q, r = divrem(num, den); @assert iszero(r)
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return q
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end
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```
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which gives
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* `35, 3003, 383724, 58790875, 10011037452, 1827174287820, 350280152218800, 69656361253789275, 14250522671900707500, 2982164406170216424300, 635707916954453388942000, 137613009450274251664451148`, resp.
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* `32928, 4671810, 759230472, 134282273216, 25166658696000, 4919891369426550, 993186502108515000, 205625998084534750800, 43449470935521085094400, 9336731949069856461585000, 2034842637042393404135380128, 448826044126481544919237242240`.

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