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Decomposition matrices and blocks for the symplectic blob algebra over the complex field

Oliver H. King, Paul P. Martin, Alison E. Parker

Abstract

The symplectic blob algebra is a physically motivated quotient of the Hecke algebra $H(\tilde{C}_n)$ with a diagram calculus. We find the blocks for the symplectic blob algebra for all specialisations of its parameters over the complex numbers. We determine Gram determinants for the cell modules with respect to a canonical contravariant form. We show in particular that the algebra is semisimple over the complex numbers unless at least one of the "quantisation" parameters, or the sum or difference of two of these parameters is integral, or the bulk parameter $q$ is a root of unity. We find decomposition numbers in many of the $q$-generic cases.

Decomposition matrices and blocks for the symplectic blob algebra over the complex field

Abstract

The symplectic blob algebra is a physically motivated quotient of the Hecke algebra with a diagram calculus. We find the blocks for the symplectic blob algebra for all specialisations of its parameters over the complex numbers. We determine Gram determinants for the cell modules with respect to a canonical contravariant form. We show in particular that the algebra is semisimple over the complex numbers unless at least one of the "quantisation" parameters, or the sum or difference of two of these parameters is integral, or the bulk parameter is a root of unity. We find decomposition numbers in many of the -generic cases.

Paper Structure

This paper contains 26 sections, 37 theorems, 92 equations, 22 figures, 4 tables.

Table of Contents

  1. Notation and preliminary definitions
  2. Review of construction of $b^{x}_n$ cell modules
  3. On standard and De Gier--Nichols weight labelling
  4. On $\pmb{\delta}$ parameter conventions and reparameterisation
  5. Ground ring arithmetic
  6. Parameterisation by exponents $w_1, w_2$
  7. Bases of the $b^{x}_n$-module $W^{n}_{}(b) = S_n(0)$
  8. Path basis for $W^{n}_{}(b)$ for generic $\pmb{\delta}$
  9. Restricting standard modules to the blob algebra
  10. A necessary block condition
  11. The central element $Z_n$
  12. Aside on substitutions
  13. The $Z_n$-action theorem
  14. Gram determinants for cell modules
  15. Homological tools for decomposition matrices and blocks of $b'_n$
  16. ...and 11 more sections

Key Result

Proposition 1.2

The ideals of $b^{x}_n$ generated by the elements $d_l$ in Figure fig:cmwl2 include according to the poset structure indicated by the Hasse diagram in Figure fig:cmwl. ∎

Figures (22)

  • Figure 1: Generating diagrams for the symplectic blob algebra.
  • Figure 2: Graphical depiction of morphisms and reflection orbits for the cell modules of $b^x_{13}$ with quantisation parameters $w_1=3$ and $w_2=1$.
  • Figure 3: (a) Cell-module weight-label poset. Here $0$ is the maximal element and $-n$ is the minimal element. (b) Cell-module weight label poset in DN-labelling (even $n=2m$ case), see §\ref{['sec:dnlabel']}.
  • Figure 4: Representative diagrams in the cell ideal poset.
  • Figure 5: Half-diagram basis of cell module ${S}_{5}(1) = W^{(5,2)}_{-,-}$.
  • ...and 17 more figures

Theorems & Definitions (57)

  • Definition 1.1
  • Proposition 1.2: gensymp
  • Proposition 2.1: gensymp
  • Definition 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Remark
  • Theorem 4.5: degiernichols
  • Proposition 5.1: gensymp
  • ...and 47 more