Conjugacy classes and automorphisms of twin groups

Abstract

The twin group {T_{n}} is a right-angled Coxeter group generated by {n-1} involutions, and the pure twin group {\mathrm{PT}_{n}} is the kernel of the natural surjection from {T_{n}} onto the symmetric group on n symbols. In this paper, we investigate some structural aspects of these groups. We derive a formula for the number of conjugacy classes of involutions in {T_{n}}, which, quite interestingly, is related to the well-known Fibonacci sequence. We also derive a recursive formula for the number of z-classes of involutions in {T_{n}}. We give a new proof of the structure of {\operatorname{Aut}(T_{n})} for {n\geq 3}, and show that {T_{n}} is isomorphic to a subgroup of {\operatorname{Aut}(\mathrm{PT}_{n})} for {n\geq 4}. Finally, we construct a representation of {T_{n}} to {\operatorname{Aut}(F_{n})} for {n\geq 2}.

Beyond braid statistics: Constructing a lattice model for anyons with exchange statistics intrinsic to one dimension

“…This hints at an effectively induced attractive interaction, something previously noted for the braid-anyon-Hubbard model [30]. In the next subsection, where we investigate (14,21) on 20 lattice sites without on-site interaction (U = 0), obtained from diagonalizing the single-particle density matrix 〈t † i t j 〉 with the traid anyon operators defined in Eqs. 17 and 22.…”

“…14 and 17a, Figs. 10(e,f) additionally show two different nontrivial abelian representations for 4 particles, namely (+ − −) and (+ + −), defined by the generalized traid anyon lattice model (21,22). In analogy to the ground state densities and energies discussed above, the densities of the natural orbitals corresponding to mirrored abelian representations [e.g., (+−) and (−+)] are also mirrored with identical occupation numbers.…”

“…4, we do not construct the polynomial explicitly and only set the desired inputs/outputs. Analogously, we can find a Jordan-Wigner-like transformation that brings the generalized Hamiltonian of the traid anyon model (21) to the form (11) with a quadratic nearest neighbor tunneling term:…”

“…The top left plot depicts the trivial case of bosons and pseudofermions, where all exchange phases are identical. The bottom two plots show 4 representations which can only be realized by the generalized version of our lattice model (21).…”

“…Figure10: Ground state natural orbitals (insets) and their occupation numbers (main plots) for our traid anyon lattice model(14,21) on 20 lattice sites without on-site interaction (U = 0), obtained from diagonalizing the single-particle density matrix 〈t † i t j 〉 with the traid anyon operators defined in Eqs. 17 and 22.…”