Hecke Symmetries and Generic Quantum Schur–Weyl Duality

The bridge starts with an explicit operator rather than an assumed universal R-matrix. A Hecke symmetry is an invertible Yang–Baxter operator satisfying a specified quadratic relation; the local relation checks construct an action. Quantum Schur–Weyl duality is a stronger equality of commutants and receives its own proof, parameter assumptions and foundational closure.

This is a prose scaffold for future item authoring. The constructions and results below are explicit proof obligations; an empty item list does not certify that they have been proved in the library. The source-grounded contracts and prerequisite audit are recorded in research/plan-hopf-hecke-algebras-track.md.

Construction and proof obligations

def-hh-hecke-symmetry-and-ordered-basis-operator. On ordered basis e_a define T(e_a⊗e_a)=q e_a⊗e_a; for a<b flip; for a>b flip+(q−q^-1)e_a⊗e_b. Define Hecke symmetry by invertibility, local braid equation and (T−q)(T+q^-1)=0. This definition is a property; the next lemma proves the model has it.

lem-hh-type-a-tensor-operator-is-a-hecke-symmetry. On each two-dimensional ordered-pair span prove the quadratic and inverse T^-1=T−(q−q^-1). Check adjacent braid equality in all equality/order patterns for triples, displaying a complete finite-case computation. Distant commutation is separate. Convert to S=qT,Q=q² explicitly.

thm-hh-hecke-symmetry-tensor-power-action. Use HH-12 presentation universal property to obtain Hn→End(V^⊗n). A quasitriangular braiding factors this way only after an extra quadratic relation is checked. At q=1 recover the permutation action; no abstract algebra-image identification in small dimension.

def-hh-type-a-quantum-tensor-action-for-centralizers. Over F=Q(q) use commuting invertible L_j (1≤j≤N), E_i,F_i (i<N) and K_i=L_i L_(i+1)^−1. Specify L_j E_i L_j^−1=q^(δ_ji−δ_j,i+1)E_i, the inverse exponent for F_i, [E_i,F_j]=δ_ij(K_i−K_i^−1)/(q−q^−1), distant E/E and F/F commutation, and adjacent Serre E_i²E_j−(q+q^−1)E_iE_jE_i+E_jE_i²=0 and its F counterpart. Define the associative presented algebra using HH-1. Candidate vector actions are L_j e_a=q^δ_ja e_a, E_i e_a=δ_(a,i+1)e_i, F_i e_a=δ_(a,i)e_(i+1), with ΔE_i=E_i⊗K_i+1⊗E_i, ΔF_i=F_i⊗1+K_i^−1⊗F_i, ΔL_j=L_j⊗L_j. The next relation-descent lemma proves these give the promised tensor actions; global quantum PBW is unnecessary.

lem-hh-quantum-presentation-and-tensor-action-descent. Check vector matrices satisfy every presented relation, then establish coproduct descent on those relations: Cartan relations follow by moving K past E/F, and the commutator splits into its two factor commutators with mixed terms cancelling. For adjacent Serre, write ΔE_i=A+B with A=E_i⊗K_i, B=1⊗E_i and AB=q²BA; expand the cubic Serre expression, group terms by degree in each tensor factor, and cancel the mixed bidegrees using K_iE_j=q^(−1)E_jK_i and the coefficients 1,−(q+q^−1),1. Pure terms are the factor Serre relations; the F calculation uses inverse Cartan weights. Distant relations use commuting factors. Set ε(L)=1, ε(E)=ε(F)=0 and reversing S(L)=L^−1, S(E)=−E K^−1, S(F)=−K F; check the same relations descend and both antipode equations on generators. Iterating Δ proves all tensor formulas satisfy all relations, with finite Laurent matrix entries. This verifies existence/nontriviality and coproduct convention before any commutation or centralizer theorem.

lem-hh-quantum-and-hecke-tensor-actions-commute. For each j compare T(L_j⊗L_j) and (L_j⊗L_j)T on a basis pair: both are the same weight scalar times T. For E_i compare T(E_i⊗K_i+1⊗E_i) with its reverse product on all pairs (a,b), partitioned by whether each label is i, i+1 or an outside label and by its order. The only two-term contribution is the (i+1,i+1) pair, where equality is q=(q−q^−1)+q^−1; the mixed i/i+1 pairs and outside-label pairs follow the displayed coefficients without omitted cases. Apply the same complete partition to F_i⊗1+K_i^−1⊗F_i. On tensor powers, the two summands touching neighboring sites form the checked two-site coproduct; every other term is identity or K_i⊗K_i there. This proves commuting inclusion for all generators and powers, before commutant equality.

lem-hh-classical-tensor-centralizer-by-polarization. Over Q, identify End(V^⊗n) with End(V)^⊗n; permutation conjugation permutes factors, so its commutant is spanned by orbit sums. Expanding Σ_(J⊆[n])(−1)^(n−|J|)(Σ_(j∈J)a_j)^⊗n gives Σ_(σ∈S_n)a_(σ1)⊗⋯⊗a_(σn); divide by repeated-label stabilizers to span every orbit sum by a^⊗n. For x_b equal to a on tensor position b, the commuting power sums p_m=Σx_b^m are classical Lie tensor operators of a^m. Derive Newton recursion m e_m=Σ_(j=1)^m(−1)^(j−1)e_(m−j)p_j by differentiating ∏(1+x_b t) as a formal coefficient identity. Then a^⊗n=e_n belongs to the classical Lie image. Adjacent E_i,F_i and diagonal counts generate all matrix Lie operators by commutators. This proves A_0=End_(B_0) without Young symmetrizers, highest weights or a hidden irreducibility exercise.

lem-hh-finite-semisimple-matrix-double-commutant. Work entirely over Q. Average finite projections for a finite group to prove Maschke complete reducibility. For each simple S, D=End_B(S) is a division algebra by kernel/image; view S as a right D^op-space. Prove finite density using a D^op-basis v_1,…,v_d: the evaluation image of B→S^d is a submodule. If proper, finite semisimplicity supplies a nonzero B-map S^d→S killing it, necessarily Σd_j on the coordinates with d_j∈D. Evaluating at 1 gives Σd_j(v_j)=0, contradicting D^op-independence. Thus B maps onto End_(D^op)(S). Apply the same argument to a sum of such tuples for distinct simples; intertwiners between distinct simples are zero, so the image is the direct sum of the independent matrix blocks. On W=⊕S_λ^(m_λ), End_B(W)=⊕Mat_(m_λ)(D_λ); commuting with diagonal and off-diagonal matrix units forces its commutant to be the original blocks End_(D_λ^op)(S_λ) acting equally on the multiplicity copies. This proves the finite semisimple bicommutant without algebraic closure, FTA, splitness or generic quantum semisimplicity. Finite composition and complement facts were supplied in HH-1.

lem-hh-finite-matrix-rank-under-specialization. For a finite matrix over R=Q[q,q^−1]_(q−1), any nonzero minor at q=1 is nonzero in F=Q(q), so generic rank is at least specialized rank. Therefore kernel dimension can only decrease. Apply this separately to vectorized word-column matrices (image dimension lower bounds) and vectorized commutator matrices (commutant dimension upper bounds). A finite word basis exists because an increasing sequence of matrix-word spans stabilizes within the finite endomorphism dimension. Only selected regular words are specialized; arbitrary rational denominators are not assumed regular.

lem-hh-classical-and-generic-tensor-double-centralizer-suppliers. Combine the preceding classical polarization/Newton lemma and finite semisimple double-commutant lemma to obtain A_0=End_(B_0) and B_0=End_(A_0) over Q. Set R=Q[q,q^−1](q−1); every tensor matrix is regular there. Add H_j=(L_j−1)/(q−1), whose diagonal entry on a word with m copies of j is 1+q+⋯+q^(m−1) (zero for m=0); these are in A_F and specialize to all classical diagonal counts. Form two finite vectorized commutator systems, one for the Hecke T_a, one for E_i,F_i,L_j^±1,H_j. The rank-specialization lemma gives dim End(B_F)≤dim A_0 and dim End_(A_F)≤dim B_0. Lift finite word bases of A_0 and B_0 to regular matrices; their nonzero minors give dim A_F≥dim A_0 and dim B_F≥dim B_0. The proved commuting inclusions give the reverse inequalities, so both sandwiches are equalities. No generic quantum semisimplicity or character theorem is imported.

thm-hh-generic-quantum-schur-weyl-mutual-centralizers. For F=Q(q), N≥1 and n≥0, the two proved rank sandwiches imply A_F=End_(B_F)(V^⊗n) and B_F=End_(A_F)(V^⊗n), where A_F and B_F are the quantum and Hecke images. This includes N<n without identifying the Hecke image with the abstract algebra. When N≥n, the Hecke orbit of e_1⊗⋯⊗e_n specializes to the n! distinct permutation tensors; a nonzero minor and HH-12 basis prove faithfulness generically. n=0 and N=1 give scalar images and scalar commutants directly. Root-of-unity and modular equalities are outside this theorem.

Reading and applications

Prerequisite pages: tensor-coherence-and-algebraic-descent, bialgebras-convolution-and-antipode-identities, hopf-module-tensor-products-and-rigid-duality, quasitriangular-hopf-algebras-and-braided-module-categories, coxeter-presentations-exchange-and-reduced-word-theorems, generic-coxeter-hecke-algebras-and-the-standard-basis, hecke-base-change-semisimplicity-and-deformation. The companion hecke-symmetries-and-generic-quantum-schur-weyl-duality-examples develops the calculations and failures needed to test these constructions.