SECANT VARIETIES OF Gr(4,8): FOUR QUBITS AND REFLECTION GROUPS

The fourth secant variety of the Grassmannian Gr(4,8) in P(Λ^4 C^8) = P^69 is defective: it has dimension 63 instead ofthe expected 67. Splitting C^8 into four planes, we show that its affine cone is the closure of the SL_8-orbit of thespace (C^2)^{⊗4} of four-qubit states. This space is the fixed space of a three-dimensional torus, which is the identitycomponent of the generic stabilizer. Main results:- The ideal of σ_4(Gr(4,8)) contains no equation of degree ≤ 8.- Its degree-9 part is a single copy of Λ^4 C^8, spanned by the 70 partial derivatives of the invariant equation R_10 of degree 10.- In degree 10 the ideal contains a 4837-dimensional module, Δ*·∇R_10 ≅ (Δ⊗Δ)/sl_8.- The invariant part of the ideal is the complete intersection (R_10, R_14, R_18). The restriction of the invariants of the θ-representation Λ^4 C^8 of E_7 to the secant variety is the polynomial ring in the invariants of degrees 2, 6, 8, 12. In the coefficients of R_18, we correct three misprints in Chen, Đoković, Grassl and Zeng (Phys. Rev. A 88, 2013), arXiv:1309.0791v1.- For r = 2, 3, 4, 5 the quotient of the cone over σ_r by SL_8 is F_r/W_r. Here F_r is a subspace of a Cartan subspace and W_r is a reflection group of type A_1, G_2, F_4, E_7. For r ≤ 4, W_r is the relative Weyl group of a parabolic subgroup of W(E_7) of type D_6, A_5'', 3A_1''. It is also the full automorphism group of the root system of the first factor sl_2, sl_3, so_8 of a regular maximal-rank subalgebra of e_7, whose odd part is the model of σ_r.- A θ-group version of an observation of Kostant: for r = 2, 3, 4 the gradient of every invariant equation of σ_r vanishes on σ_r. This explains the first equations of σ_4, and an equation of degree 7 of σ_3.- σ_4(Gr(4,8)) is an irreducible component of the singular locus of the hypersurface R_10 = 0. The scheme V(∇R_10) is smooth, hence reduced, at its general point. The proofs combine three ingredients:- geometric and Lie-theoretic arguments;- exact computations over Q;- certificates modulo p = 2^31 − 1, used only as lower bounds for ranks over Q. All computations are reproducible with the supplementary material, which needs only Python 3 and numpy. FILES IN THIS RECORD- sigma4_G48.pdf: the paper.- sigma4_G48_supplementary.zip: the supplementary material. It contains MANIFEST.txt (every file with its sha256) and two folders, which must be kept side by side. Each folder has a README.txt that says what every script proves and in which statement of the paper it is used. - sigma4_G48_probe/ (44 files): - the decompositions of S^d(Λ^4 C^8) for d ≤ 10 (charA.py and its outputs); - the extremal projector (Asherova–Smirnov–Tolstoy) and its checkpointed, vectorized and memory-lean variants, with tests showing that they compute the same numbers; - the certificate logs for degrees ≤ 9, with the degree-8 run made with a recorded seed; - the Hodge-duality coverage check for degree 9; - the covariant bound of Section 6. - sigma4_G48_invariants/ (46 files): - exact computations of Katanova's invariants on the four-qubit family and on σ_2; - the model checks; - the reflection groups W(G_2) and W(F_4); - the e_7 root and centralizer computations; - the gradient, Hessian and degree-10 computations; - the comparison with Chen–Đoković–Grassl–Zeng; - an independent implementation of the invariants that shares no code with the other scripts.- MANIFEST.txt: a copy of the file list with checksums, for preview. HOW TO REPRODUCEUnzip, enter a folder and run the scripts with python3, in the order given in its README.txt. Most scripts run inseconds to about two minutes. syz10.py takes about 16 minutes. The full degree-9 projector runs take hours; they arecheckpointed and can be resumed. RELATED WORKCompanion paper on the third secant variety:J. Torregrosa Jiménez, "Binary quartics and the equations of the third secant variety of the Grassmannian Gr(4,8)",Zenodo, DOI 10.5281/zenodo.23916400.