The same character table, and still not isomorphic – the same has four strengths, and there is one more rung below them.
The corpus has used the word same many times: the same root (Paper 142), the same second cohomology (Paper 147), the same cover (Paper 151). This paper shows that the word has four strengths and places one example below all of them. NO NEW MATHEMATICAL THEOREM IS CLAIMED. That the dihedral group of the square and the quaternion group share a character table, the Frobenius-Schur indicator, Morita equivalence, and the distinction between equivalence and isomorphism of categories are all standard.
First, two groups can have identical character tables and fail to be isomorphic. The symmetry group of the square and the quaternion group both have order eight and five conjugacy classes of sizes one, one, two, two and two. In both the commutator subgroup has order two, so the abelianisation is the Klein four-group and there are four linear irreducibles; since the squares of the degrees sum to eight, the remaining irreducible has degree two and is the defining representation. Its character takes the value two at the identity, minus two at the central involution and zero on the other three classes – identical in both cases. The tables therefore coincide. Yet the symmetry group of the square has five elements of order two and the quaternion group has one (all computed here). An isomorphism preserves orders, so the two groups are not isomorphic. Everything built from characters – degrees, the number of classes, the decomposition of the group algebra, multiplicities – agrees, and the objects still differ.
Second, the separator is the Frobenius-Schur indicator. Computing the sum, over the group and divided by its order, of the character evaluated at squares gives plus one for the symmetry group of the square and minus one for the quaternion group (computed here): real type and quaternionic type. Why this quantity cannot be built from the character table is plain: it uses the operation of squaring an element. A character table is a function on conjugacy classes and does not remember how squaring moves the classes. That is why one number sees a difference the table cannot.
Third, as a consequence the quaternion group’s two-dimensional representation has no real form. The real two-by-two matrices squaring to minus the identity are exactly those with opposite diagonal entries and with the sum of the square of that entry and the product of the off-diagonal entries equal to minus one. Sampling four hundred thousand random pairs from that set and measuring the minimum of the absolute trace of the product gave 2.000003, never reaching the zero that anticommutation requires (computed here); the exact lower bound is likewise two. The reason lies in a signature. On trace-free real two-by-two matrices the form given by half the trace of the square has signature two and one. Squaring to minus the identity says that the self-inner-product is minus one, and anticommuting says that the mutual inner product is zero, so two orthogonal vectors of negative norm would be needed – impossible in a space with only one negative direction. Over the complex numbers it is attainable, and the quaternion matrices attain it, their product having trace zero. The quaternion group’s two-dimensional representation therefore needs the quaternions while the other needs only real matrices, and the sign of the Frobenius-Schur indicator is exactly that question – over which field can it be written – turned into a number.
Fourth, equivalence is not isomorphism. The ring of integers and the ring of two-by-two integer matrices are not isomorphic: the latter is non-commutative and has a non-zero element whose square is zero, while the integers have no such element (checked here). Their module categories are nevertheless equivalent, which is Morita equivalence. What survives: the centre, the lattice of two-sided ideals, K-theory. What does not: commutativity, and the ring itself. Solving directly for the centre of the matrix ring gives exactly the scalar matrices, isomorphic to the integers. It is precisely because the centre survives that Morita equivalence collapses to isomorphism among commutative rings. How far the relation collapses depends on what the objects are.
Fifth, the same thing happens at the level of categories. The category of finite-dimensional vector spaces has a proper class of objects, while its skeleton has one object per dimension and is countable. The two are equivalent and not isomorphic, since an isomorphism of categories would require a bijection on objects. Equivalence therefore does not preserve even how many objects there are, and saying the same category says nothing until one writes isomorphic or equivalent.
The census comes out as follows. Equal – being the same object. Isomorphic – there being an invertible map. Equivalent – being invertible up to isomorphism. The same root – having arrived from one place. And below all four there is a rung on which invariants merely agree, and that is where the two groups of this paper sit. Paper 142 wrote that a shared root is not identity; this paper shows there is another rung beneath it. Paper 110’s separation of the string beta function equalling the Ricci curvature as gold from the surgery-renormalisation resemblance as a rhyme was a descent of the same staircase.
IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new mathematical theorem (there is none). What is added: arranging the corpus’s uses of the word same into four strengths, placing beneath them an example in which every invariant agrees and the objects still differ, and computing the separator as a single number. What remains: seen through the strongest tool of the subject, its characters, the two groups were indistinguishable. What told them apart was one number that remembers squaring. They look the same says nothing until one writes down which tool did the looking – and adding one tool splits a single thing into two.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). The Same Character Table, and Still Not Isomorphic [Paper 152]. Zenodo.
DOI: 10.5281/zenodo.22084202
ORCID: 0009-0006-8587-3450