This is a follow-up to Representations with the same degree, about Frank Lübeck’s preprint and Andy Huchala’s undergraduate thesis on irreducible representations of a simple algebraic group that share their dimension. In the corrections to that post I noted that the clean construction sequences (such as A000891 in type $\mathrm{A}_l$) are not the smallest degrees at which two non-isomorphic, non-dual irreducibles coincide in dimension. This post collects the genuine smallest degrees.
For each classical type, here is the smallest degree carrying two irreducible representations of equal dimension that are not related by a diagram automorphism, for the ranks $l$ shown. They are kept as plain text so they are easy to copy into the OEIS:
A, l = 2..25: 15, 20, 70, 105, 210, 1512, 4620, 4950, 8008, 924, 52052, 80080, 928200, 495040, 271320, 3060, 5155080, 1009470, 10296594, 14805560, 20366500, 2884200, 37001250, 48976200
B, l = 2..25: 35, 112, 2772, 23595, 715, 6636630, 23984688, 66927861, 9585192960, 669442048, 251608500, 93475514149875, 2583610029, 2184324842700, 1107568, 59454145018522800, 3359934925776, 3078010375997128, 1255300994827620, 334620762580894464675, 2664821443401, 23398175753138864128, 67778385492333035520, 37701404492850988890000
D, l = 4..25: 672, 210, 352, 24024, 1067040, 2494206, 5457408, 26334, 460337701824, 14880153600, 13571131392000, 54627300, 234371284992, 30408171847680, 20123126100, 2327117992710356160, 547214951521155808800, 52569351652198318080, 34600877643532200000, 53524680, 49180016759537664000, 502382039575652375139000
These are wildly irregular — not monotone in $l$, and nothing like the smooth construction sequences (in type $\mathrm{A}$ they coincide with A000891 only at $l=3$). The minima keep dipping back down: $\mathrm{A}_{11}$ falls to $924$ from the coincidence $\dim\mathrm{V}(\omega_6)=\dim\mathrm{V}(\omega_1+2\omega_{11})$, and $\mathrm{A}_{17}$ all the way to $3060$.
A witnessing pair for each entry (there can be more than two representations at the minimal degree; two suffice), here for the first ranks $l\leq10$:
| Dynkin type | first weight | second weight | common degree |
|---|---|---|---|
| $\mathrm{A}_{2}$ | $4\omega_{2}$ | $\omega_{1}+2\omega_{2}$ | 15 |
| $\mathrm{A}_{3}$ | $3\omega_{3}$ | $\omega_{2}+\omega_{3}$ | 20 |
| $\mathrm{A}_{4}$ | $4\omega_{4}$ | $\omega_{1}+2\omega_{4}$ | 70 |
| $\mathrm{A}_{5}$ | $2\omega_{4}$ | $\omega_{3}+\omega_{5}$ | 105 |
| $\mathrm{A}_{6}$ | $4\omega_{6}$ | $\omega_{4}+\omega_{6}$ | 210 |
| $\mathrm{A}_{7}$ | $\omega_{5}+2\omega_{7}$ | $\omega_{4}+\omega_{6}$ | 1512 |
| $\mathrm{A}_{8}$ | $\omega_{5}+2\omega_{8}$ | $2\omega_{2}+\omega_{8}$ | 4620 |
| $\mathrm{A}_{9}$ | $2\omega_{8}+\omega_{9}$ | $2\omega_{7}$ | 4950 |
| $\mathrm{A}_{10}$ | $6\omega_{10}$ | $\omega_{9}+3\omega_{10}$ | 8008 |
| $\mathrm{B}_{2}$ | $4\omega_{2}$ | $\omega_{1}+2\omega_{2}$ | 35 |
| $\mathrm{B}_{3}$ | $3\omega_{3}$ | $\omega_{2}+\omega_{3}$ | 112 |
| $\mathrm{B}_{4}$ | $4\omega_{4}$ | $\omega_{2}+2\omega_{4}$ | 2772 |
| $\mathrm{B}_{5}$ | $2\omega_{4}$ | $\omega_{3}+\omega_{4}$ | 23595 |
| $\mathrm{B}_{6}$ | $\omega_{4}$ | $\omega_{1}+\omega_{2}$ | 715 |
| $\mathrm{B}_{7}$ | $\omega_{6}+2\omega_{7}$ | $\omega_{5}+2\omega_{7}$ | 6636630 |
| $\mathrm{B}_{8}$ | $2\omega_{6}$ | $\omega_{5}+\omega_{6}$ | 23984688 |
| $\mathrm{B}_{9}$ | $2\omega_{2}+\omega_{5}$ | $8\omega_{1}+\omega_{2}$ | 66927861 |
| $\mathrm{B}_{10}$ | $\omega_{2}+3\omega_{10}$ | $\omega_{2}+\omega_{7}+\omega_{10}$ | 9585192960 |
| $\mathrm{D}_{4}$ | $5\omega_{4}$ | $\omega_{3}+3\omega_{4}$ | 672 |
| $\mathrm{D}_{5}$ | $\omega_{4}+\omega_{5}$ | $3\omega_{1}$ | 210 |
| $\mathrm{D}_{6}$ | $\omega_{1}+\omega_{6}$ | $3\omega_{1}$ | 352 |
| $\mathrm{D}_{7}$ | $\omega_{1}+\omega_{5}$ | $\omega_{1}+2\omega_{2}$ | 24024 |
| $\mathrm{D}_{8}$ | $2\omega_{2}+\omega_{3}$ | $4\omega_{1}+\omega_{3}$ | 1067040 |
| $\mathrm{D}_{9}$ | $2\omega_{1}+\omega_{6}$ | $3\omega_{1}+2\omega_{2}$ | 2494206 |
| $\mathrm{D}_{10}$ | $\omega_{5}+\omega_{10}$ | $2\omega_{2}+\omega_{10}$ | 5457408 |
The table and the sequences above come from a degree-bounded backtrack that enumerates every dominant weight below a candidate degree, quotienting out the diagram automorphisms — duality in type $\mathrm{A}$, the spinor-node swap in $\mathrm{D}_l$ for $l\geq5$, and triality in $\mathrm{D}_4$:
# diagram automorphisms acting on coordinates
autos(::Type{TypeA{l}}) where l = [identity, reverse]
autos(::Type{TypeB{l}}) where l = [identity]
autos(::Type{TypeD{4}}) = [v -> v[[a,2,b,c]] for (a,b,c) in
[(1,3,4),(1,4,3),(3,1,4),(3,4,1),(4,1,3),(4,3,1)]] # triality
autos(::Type{TypeD{l}}) where l = [identity, v -> v[[1:l-2; l; l-1]]]
# bin all dominant weights of degree <= bound by degree (pruning: degree is
# increasing in each coordinate, so trailing zeros give a lower bound)
function weights(::Type{T}, bound) where T <: DynkinType
l = rank(T); bins = Dict{BigInt,Vector{Vector{Int}}}(); v = zeros(Int, l)
rec(i) = i > l ? push!(get!(() -> Vector{Int}[], bins, degree(T, v)), copy(v)) :
(while degree(T, v) <= bound; rec(i + 1); v[i] += 1 end; v[i] = 0)
rec(1); bins
end
# smallest degree shared by two weights in distinct diagram-automorphism orbits;
# the starting bound is only a guess — we keep growing it until a duplicate appears
function smallest_duplicate(::Type{T}) where T <: DynkinType
G = autos(T); bound = big(2000)
while true
d = [k for (k, ws) in weights(T, bound)
if length(unique(minimum(g(w) for g in G) for w in ws)) > 1]
isempty(d) || return minimum(d)
bound *= 4
end
end
[smallest_duplicate(TypeA{l}) for l in 2:25]
[smallest_duplicate(TypeB{l}) for l in 2:25]
[smallest_duplicate(TypeD{l}) for l in 4:25]
Computing all three lists up to $l=25$ takes about a minute (≈56 s on my laptop, after compilation); it stays this fast because in high rank only few dominant weights have small dimension, so even the largest bounds — some entries exceed $10^{23}$ — still enclose comparatively few weights to enumerate.