THE ANTISYMMETRY swap two inputs and the sign flips — the symmetry that forbids sameness
Some functions are symmetric under a swap; others are antisymmetric — exchange two arguments and the output negates: f(…,a,…,b,…) = −f(…,b,…,a,…). The determinant is the archetype: swap two rows and it flips sign — which forces it to vanish when two rows are equal (a thing equal to its own negative is zero). That is why a determinant detects linear dependence, and why the Pauli exclusion principle follows from an antisymmetric wavefunction: two identical fermions in the same state give ψ=−ψ=0 — forbidden. Antisymmetry is the symmetry that rules out coincidence.
THE TECHNIQUE swap two arguments ⇒ output negates; equal arguments ⇒ zero
The demo swaps the rows of a 2×2 and watches the determinant flip sign: live demo
HISTORY & CREDIT Grassmann · Pauli 1925
“Swapping inputs is harmless.” — for an antisymmetric function it flips the sign, and forces the output to vanish when two inputs coincide. The symmetry that forbids sameness. cited
the relation · odd under a transposition; equal arguments force 0. the determinant · the alternating multilinear form (Grassmann's exterior algebra); zero iff rows are dependent. 1925 · Wolfgang Pauli — the exclusion principle: fermion wavefunctions are antisymmetric, so no two share a state.
Exchange two inputs, flip the sign; make them equal, get zero — the symmetry that forbids two things from being the same. Grassmann / Pauli
RECOMMEND FOR I-13 the sign-flip on swap, on the compiler
On the canonical compiler, det[[1,2],[3,4]] = −2; swap the rows and it becomes 2 = −(−2):
$ i13 run sy_antisymmetry.i13 # swap rows of a 2x2 -> det negates
RUN OK · 22 step(s) · peak stack 3 · call depth 0
det = -2 swapped = 2
antisym = 1 -- swap negates; equal rows would force det = 0
Recommend: antisymmetry is the symmetry that forbids coincidence — swap negates, so equal inputs vanish — and i13 flips the determinant's sign. Not a keeper (odd-under-swap is a symmetry a function has, witnessed; and the negate-on-swap is a self-inverse flavour). The dart that shows one sign convention rules the exclusion principle: two identical fermions give ψ=−ψ=0, so matter takes up space.