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The linear-independence conjecture for minimal intermediate subfactors

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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 1Other wordings

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).

References

Primary source

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

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