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The linear-independence conjecture for minimal intermediate subfactors

Let N⊂MN\subset M be an irreducible subfactor with finite Jones index, and let PiP_i, 1≤i≤n1\leq i\leq n, be the minimal intermediate subfactors. Let M1M_1 be the Jones basic construction, let ei∈N′∩M1e_i\in N'\cap M_1 be the Jones projection from MM onto PiP_i, and let eNe_N be the Jones projection from MM onto NN.

Linear-independence conjecture. There exist vectors ξi,ξ∈N′∩M1\xi_i,\xi\in N'\cap M_1 such that

eiξi=ξi(1≤i≤n),eNξ=ξ,e_i\xi_i=\xi_i\quad(1\leq i\leq n),\qquad e_N\xi=\xi,

and the vectors ξ1,…,ξn,ξ\xi_1,\ldots,\xi_n,\xi are linearly independent.

The conjecture is a strengthening of the minimal-intermediate-subfactor form of Wall's conjecture and uses the algebra structure of N′∩M1N'\cap M_1. The source proves it for fixed-point subfactors RG⊂RR^G\subset R when GG is solvable, while noting that it does not automatically imply the dual or maximal version.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The linear-independence conjecture for minimal intermediate subfactors

    Let N⊂MN\subset M be an irreducible subfactor with finite Jones index. Let PiP_i, 1≤i≤n1\leq i\leq n, be the minimal intermediate subfactors. For each ii, let ei∈N′∩M1e_i\in N'\cap M_1 be the Jones projection from MM onto PiP_i, and let eNe_N be the Jones projection from MM onto NN.

    Linear-independence conjecture. There exist vectors ξi,ξ∈N′∩M1\xi_i,\xi\in N'\cap M_1 such that

    eiξi=ξi(1≤i≤n),eNξ=ξ,e_i\xi_i=\xi_i\quad(1\leq i\leq n),\qquad e_N\xi=\xi,

    and ξ1,…,ξn,ξ\xi_1,\ldots,\xi_n,\xi are linearly independent.

    The source identifies this as a conjecture formulated in earlier work and notes that it implies the subfactor version of Wall's conjecture. Its general status is not resolved in the supplied text.

    source: Robert Guralnick and Feng Xu, “On a subfactor generalization of Wall's conjecture”, arXiv:1006.5947 (2010).

Sources & referencesView supporting material

Primary source

Feng Xu, “On intermediate subfactors of Goodman-de la Harpe-Jones subfactors”, arXiv:1002.2744 (2010).

operator-algebrassubfactors

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