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Different Faces of the Shearlet Group

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Abstract

Recently, shearlet groups have received much attention in connection with shearlet transforms applied for orientation sensitive image analysis and restoration. The square integrable representations of the shearlet groups provide not only the basis for the shearlet transforms but also for a very natural definition of scales of smoothness spaces, called shearlet coorbit spaces. The aim of this paper is twofold: first we discover isomorphisms between shearlet groups and other well-known groups, namely extended Heisenberg groups and subgroups of the symplectic group. Interestingly, the connected shearlet group with positive dilations has an isomorphic copy in the symplectic group, while this is not true for the full shearlet group with all nonzero dilations. Indeed we prove the general result that there exist, up to adjoint action of the symplectic group, only one embedding of the extended Heisenberg algebra into the Lie algebra of the symplectic group. Having understood the various group isomorphisms it is natural to ask for the relations between coorbit spaces of isomorphic groups with equivalent representations. These connections are examined in the second part of the paper. We describe how isomorphic groups with equivalent representations lead to isomorphic coorbit spaces. In particular we apply this result to square integrable representations of the connected shearlet groups and metaplectic representations of subgroups of the symplectic group. This implies the definition of metaplectic coorbit spaces. Besides the usual full and connected shearlet groups we also deal with Toeplitz shearlet groups.

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Acknowledgments

This work has been supported by Deutsche Forschungsgemeinschaft (DFG), Grants DA 360/19–1 and STE 571/11–1. Some parts of the paper have been written during a stay at the Erwin-Schrödinger Institute (ESI), Vienna, Workshop on “Time-Frequency Analysis”, January 13–17 2014. Therefore the support of ESI is also acknowledged. F. De Mari and E. De Vito were partially supported by Progetto PRIN 2010–2011 “Varietà reali e complesse: geometria, topologia e analisi armonica”. They are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). S. Dahlke, S. Häuser, G. Steidl and G. Teschke were partially supported by DAAD Project 57056121, “Hochschuldialog mit Südeuropa 2013”.

Author information

Authors and Affiliations

  1. FB12 Mathematik und Informatik, Philipps-Universität Marburg, Hans-Meerwein Straße, Lahnberge, 35032, Marburg, Germany

    Stephan Dahlke

  2. Dipartimento di Matematica, Università degli Studi di Genova, Via Dodecaneso 35, 16146, Genova, Italy

    Filippo De Mari & Ernesto De Vito

  3. Fachbereich für Mathematik, Technische Universität Kaiserslautern, Paul-Ehrlich-Str. 31, 67663, Kaiserslautern, Germany

    Sören Häuser & Gabriele Steidl

  4. Institute for Computational Mathematics in Science and Technology, Hochschule Neubrandenburg, University of Applied Sciences, Brodaer Str. 2, 17033, Neubrandenburg, Germany

    Gerd Teschke

Authors
  1. Stephan Dahlke
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  2. Filippo De Mari
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  3. Ernesto De Vito
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  4. Sören Häuser
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  5. Gabriele Steidl
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  6. Gerd Teschke
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Corresponding author

Correspondence to Sören Häuser.

Appendix

Appendix

In the following we list the Lie brackets of the canonical matrices \(D \in {\mathcal {N}}\) with the basis matrices \(X_\nu \), \(\nu \in \triangle \), and \(H_{1,0}\), \(H_{0,1}\) from the root space decomposition of \(\mathfrak {sp}(2,\mathbb {R})\) and the matrices \(M_{\Gamma }\) defined by (29).

Case 1. For \( D_1 = a_1H_{1,0} + a_2H_{0,1} \) we obtain

$$\begin{aligned} \begin{array}{lcllcl} [D_1,X_{\alpha }]&{}=&{} (a_1-a_2)X_{\alpha }, &{} [D_1,X_{-\alpha }] &{}=&{} -(a_1-a_2)X_{-\alpha },\\ {[}D_1,X_{\beta }]&{}=&{} 2a_2X_{\beta },&{} [D_1,X_{-\beta }]&{}=&{} - 2a_2X_{-\beta },\\ {[}D_1,X_{\alpha +\beta }]&{}=&{} (a_1+a_2)X_{\alpha +\beta }, &{} [D_1,X_{-\alpha -\beta }]&{}=&{} -(a_1+a_2)X_{-\alpha -\beta },\\ {[}D_1,X_{2\alpha +\beta }]&{}=&{} 2a_1X_{2\alpha +\beta },&{} [D_1,X_{-2\alpha -\beta }]&{}=&{} -2a_1X_{-2\alpha -\beta },\\ {[}D_1,H_{1,0}]&{}=&{} 0,&{} [D_1,H_{0,1}]&{}=&{} 0 \end{array} \end{aligned}$$

and \(M_\Gamma \) is a diagonal matrix with entries

$$\begin{aligned} \left( a_1 - a_2 - \Gamma , 2a_2 - \Gamma , a_1 + a_2 - \Gamma , 2a_1 - \Gamma , a_2 - a_1 - \Gamma , - 2a_2 - \Gamma , - a_2 - a_1 - \Gamma , - 2a_1 - \Gamma , - \Gamma , - \Gamma \right) , \end{aligned}$$

where \(a_1 \ge a_2 \ge 0\).

Case 2. For \( D_2 = X_{-\alpha } + aH_{1,0} + aH_{0,1} \) we obtain

$$\begin{aligned} \begin{array}{lcllcl} [D_2,X_{\alpha }] &{}=&{} H_{1,0}-H_{0,1}, &{} [D_2,X_{-\alpha }] &{}=&{} 0,\\ {[}D_2,X_{\beta }] &{}=&{} 2aX_{\beta }, &{} [D_2,X_{-\beta }] &{}=&{} -2aX_{-\beta }+X_{-\alpha -\beta },\\ {[}D_2,X_{\alpha +\beta }] &{}=&{} -2X_{\beta }+2aX_{\alpha +\beta }, &{} [D_2,X_{-\alpha -\beta }] &{}=&{} -2aX_{-\alpha -\beta }+X_{-2\alpha -\beta },\\ {[}D_2,X_{2\alpha +\beta }] &{}=&{} -2X_{\alpha +\beta }+2aX_{2\alpha +\beta }, &{} [D_2,X_{-2\alpha -\beta }] &{}=&{} -2aX_{-2\alpha -\beta },\\ {[}D_2,H_{1,0}] &{}=&{} X_{-\alpha }, &{} [D_2,H_{0,1}] &{}=&{} -X_{-\alpha } \end{array} \end{aligned}$$

and

$$\begin{aligned} M_\Gamma = \begin{pmatrix} -\Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 2a - \Gamma &{}\quad -2 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 2a - \Gamma &{}\quad -2 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 2a - \Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad -1 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -2a - \Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad -2a - \Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad -2a - \Gamma &{}\quad 0 &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\Gamma &{}\quad 0 \\ -1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\Gamma \end{pmatrix} \end{aligned}$$

with

$$\begin{aligned} \det \, M_\Gamma = \Gamma ^4 (\Gamma - 2a)^3 (\Gamma + 2a)^3, \qquad a \ge 0. \end{aligned}$$

Case 3. For \( D_3 = bX_{\alpha } + bX_{-\alpha } + aH_{1,0} + aH_{0,1} \) we obtain

$$\begin{aligned} \begin{array}{lcllcl} [D_3,X_{\alpha }] &{}=&{} bH_{1,0}-bH_{0,1}, &{} [D_3,X_{-\alpha }] &{}=&{} -bH_{1,0}+bH_{0,1},\\ {[}D_3,X_{\beta }] &{}=&{} 2aX_{\beta }+bX_{\alpha +\beta }, &{} [D_3,X_{-\beta }] &{}=&{} -2aX_{-\beta } + bX_{-\alpha -\beta },\\ {[}D_3,X_{\alpha +\beta }] &{}=&{} -2bX_{\beta }+2aX_{\alpha +\beta } &{}[D_3,X_{-\alpha -\beta }] &{}=&{} -2bX_{-\beta }-2aX_{-\alpha -\beta } \\ &{} &{} +bX_{2\alpha +\beta }, &{} &{} &{}+ bX_{-2\alpha -\beta },\\ {[}D_3,X_{2\alpha +\beta }] &{}=&{} -2bX_{\alpha +\beta }+2aX_{2\alpha +\beta }, &{} [D_3,X_{-2\alpha -\beta }] &{}=&{} -2bX_{-\alpha -\beta } \!-\!2aX_{-2\alpha -\beta },\\ {[}D_3,H_{1,0}] &{}=&{} -b X_{\alpha }+bX_{-\alpha }, &{} [D_3,H_{0,1}] &{}=&{} b X_{\alpha }-bX_{-\alpha } \end{array} \end{aligned}$$

and

$$\begin{aligned} M_\Gamma = \begin{pmatrix} -\Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -b &{}\quad b \\ 0 &{}\quad 2a - \Gamma &{}\quad -2b &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad b &{}\quad 2a - \Gamma &{}\quad -2b &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad b &{}\quad 2a - \Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad b &{}\quad -b \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -2a - \Gamma &{}\quad -2b &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad b &{}\quad -2a - \Gamma &{}\quad -2b &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad b &{}\quad -2a - \Gamma &{}\quad 0 &{}\quad 0 \\ b &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -b &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\Gamma &{}\quad 0 \\ -b &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad b &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\Gamma \end{pmatrix} \end{aligned}$$

with

$$\begin{aligned} \det \, M_\Gamma= & {} \Gamma ^2 (\Gamma ^2 + 4 b^2) (\Gamma - 2a) (\Gamma + 2a) \left( (\Gamma - 2 a)^2 + 4 b^2 \right) \\&\left( (\Gamma + +2 a)^2 + 4 b^2 \right) , \quad a,b>0. \end{aligned}$$

Case 4. For \( D_4 = \varepsilon X_{\alpha +\beta } - X_{-\alpha } - \frac{\varepsilon }{2}X_{-2\alpha -\beta } \) we obtain

$$\begin{aligned} \begin{array}{lcllcl} [D_4,X_{\alpha }] &{}=&{} -\varepsilon X_{2\alpha +\beta } -\varepsilon X_{-\alpha -\beta } &{} [D_4,X_{-\alpha }] &{}=&{} 2\varepsilon X_{\beta }, \\ &{} &{}-H_{1,0} +H_{0,1}, &{}&{}&{}\\ {[}D_4,X_{\beta }] &{}=&{} 0, &{} [D_4,X_{-\beta }] &{}=&{} -\varepsilon X_{\alpha } -X_{-\alpha -\beta }, \\ {[}D_4,X_{\alpha +\beta }] &{}=&{} 2X_{\beta } +\varepsilon X_{-\alpha }, &{} [D_4,X_{-\alpha -\beta }] &{}=&{} -X_{-2\alpha -\beta } -\varepsilon H_{1,0} \\ &{}&{}&{}&{}&{} -\varepsilon H_{0,1}, \\ {[}D_4,X_{2\alpha +\beta }] &{}=&{} 2X_{\alpha +\beta } -2\varepsilon H_{1,0}, &{} [D_4,X_{-2\alpha -\beta }] &{}=&{} 2\varepsilon X_{-\alpha }, \\ {[}D_4,H_{1,0}] &{}=&{} -\varepsilon X_{\alpha +\beta } -X_{-\alpha } -\varepsilon X_{-2\alpha -\beta }, &{} [D_4,H_{0,1}] &{}=&{} -\varepsilon X_{\alpha +\beta } +X_{-\alpha } \end{array} \end{aligned}$$

and

$$\begin{aligned} M_\Gamma = \begin{pmatrix} -\Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\varepsilon &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad -\Gamma &{}\quad 2 &{}\quad 0 &{}\quad 2\varepsilon &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad -\Gamma &{}\quad 2 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\varepsilon &{}\quad -\varepsilon \\ -\varepsilon &{}\quad 0 &{}\quad 0 &{}\quad -\Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \varepsilon &{}\quad 0 &{}\quad -\Gamma &{}\quad 0 &{}\quad 0 &{}\quad 2\varepsilon &{}\quad -1 &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ -\varepsilon &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -1 &{}\quad -\Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -1 &{}\quad -\Gamma &{}\quad -\varepsilon &{}\quad 0 \\ -1 &{}\quad 0 &{}\quad 0 &{}\quad -2\varepsilon &{}\quad 0 &{}\quad 0 &{}\quad -\varepsilon &{}\quad 0 &{}\quad -\Gamma &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\varepsilon &{}\quad 0 &{}\quad 0 &{}\quad -\Gamma \end{pmatrix} \end{aligned}$$

with \( \det \, M_\Gamma = \Gamma ^{10}. \)

Case 5. For \( D_5 = \frac{\varepsilon }{2} X_{2\alpha +\beta } + \frac{b^2\varepsilon }{2} X_{-2\alpha -\beta } + aH_{0,1} \) we obtain

$$\begin{aligned} \begin{array}{lcllcl} [D_5,X_{\alpha }] &{}=&{} -aX_{\alpha } + b^2\varepsilon X_{-\alpha -\beta }, &{} [D_5,X_{-\alpha }] &{}= &{} \varepsilon X_{\alpha +\beta },\\ {[}D_5,X_{\beta }] &{}=&{} 2aX_{\beta }, &{} [D_5,X_{-\beta }] &{}=&{} -2aX_{-\beta },\\ {[}D_5,X_{\alpha +\beta }] &{}=&{} a X_{\alpha +\beta }-b^2\varepsilon X_{-\alpha }, &{} [D_5,X_{-\alpha -\beta }] &{}=&{} -\varepsilon X_{\alpha }-aX_{-\alpha -\beta },\\ {[}D_5,X_{2\alpha +\beta }] &{}=&{} 2b^2\varepsilon H_{1,0}, &{} [D_5,X_{-2\alpha -\beta }] &{}= &{} -2\varepsilon H_{1,0},\\ {[}D_5,H_{1,0}] &{}=&{} -\varepsilon X_{2\alpha +\beta } + b^2\varepsilon X_{-2\alpha -\beta }, &{} [D_5,H_{0,1}], &{}=&{} 0 \end{array} \end{aligned}$$

and

$$\begin{aligned} M_\Gamma = \begin{pmatrix} -a-\Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\varepsilon &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 2a-\Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad a-\Gamma &{}\quad 0 &{}\quad \varepsilon &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad -\Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\varepsilon &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad -b^2\varepsilon &{}\quad 0 &{}\quad a-\Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -2a-\Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ b^2\varepsilon &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -a-\Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\Gamma &{}\quad b^2\varepsilon &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 2b^2\varepsilon &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -2\varepsilon &{}\quad -\Gamma &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\Gamma \end{pmatrix} \end{aligned}$$

with

$$\begin{aligned} \det \, M_\Gamma= & {} \Gamma ^2 (\Gamma - 2a) (\Gamma + 2a) (\Gamma ^2 + 4b^2) \left( (\Gamma -a)^2 + b^2 \right) \\&\left( (\Gamma ^2 + a)^2 + b^2 \right) , \quad a,b \ge 0. \end{aligned}$$

Case 6. For \( D_6 = \eta X_{\beta } + \frac{\varepsilon }{2}X_{2\alpha +\beta } + b_2^2\eta X_{-\beta } + \frac{b_1^2\varepsilon }{2}X_{-2\alpha -\beta } \) we obtain

$$\begin{aligned} \begin{array}{lcllcl} [D_6,X_{\alpha }] &{}=&{} -\eta X_{\alpha +\beta } + b_1^2\varepsilon X_{-\alpha -\beta }, &{} [D_6,X_{-\alpha }] &{}=&{} \varepsilon X_{\alpha +\beta } - b_2^2\eta X_{-\alpha -\beta },\\ {[}D_6,X_{\beta }] &{}=&{} b_2^2\eta H_{0,1}, &{} [D_6,X_{-\beta }] &{}=&{} -\eta H_{0,1},\\ {[}D_6,X_{\alpha +\beta }] &{}=&{} b_2^2\eta X_{\alpha } - b_1^2\varepsilon X_{-\alpha }, &{} [D_6,X_{-\alpha -\beta }] &{}=&{} -\varepsilon X_{\alpha } + \eta X_{-\alpha },\\ {[}D_6,X_{2\alpha +\beta }] &{}=&{} 2b_1^2\varepsilon H_{1,0}, &{} [D_6,X_{-2\alpha -\beta }] &{}= &{} -2\varepsilon H_{1,0},\\ {[}D_6,H_{1,0}] &{}=&{} -\varepsilon X_{2\alpha +\beta } + b_1^2\varepsilon X_{-2\alpha -\beta }, &{} [D_6,H_{0,1}] &{}= &{} -2\eta X_{\beta } + 2b_2^2\eta X_{-\beta } \end{array} \end{aligned}$$

and

$$\begin{aligned} M_\Gamma = \begin{pmatrix} -\Gamma &{}\quad 0 &{}\quad b_2^2\eta &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\varepsilon &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad -\Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -2\eta \\ -\eta &{}\quad 0 &{}\quad -\Gamma &{}\quad 0 &{}\quad \varepsilon &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad -\Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\varepsilon &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad -b_1^2\varepsilon &{}\quad 0 &{}\quad -\Gamma &{}\quad 0 &{}\quad \eta &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 2b_2^2\eta \\ b_1^2\varepsilon &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -b_2^2\eta &{}\quad 0 &{}\quad -\Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\Gamma &{}\quad b_1^2\varepsilon &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 2b_1^2\varepsilon &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -2\varepsilon &{}\quad -\Gamma &{}\quad 0 \\ 0 &{}\quad b_2^2\eta &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\eta &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\Gamma \end{pmatrix} \end{aligned}$$

with

$$\begin{aligned} \det \, M_\Gamma= & {} \Gamma ^2 (\Gamma ^2 + 4b_2^2) (\Gamma ^2 + 4 b_1^2) \left( \Gamma ^2 + (b_1 - b_2)^2 \right) \\&\left( \Gamma ^2 + (b_1 + b_2)^2 \right) , \quad b_1 \ge b_2 \ge 0. \end{aligned}$$

Case 7. For \( D_7 = - X_{\alpha } - \varepsilon X_{\beta } + \frac{\varepsilon }{2b^2}X_{2\alpha +\beta } - b^2X_{-\alpha } \) and

$$\begin{aligned} \begin{array}{lcllcl} [D_7,X_{\alpha }] &{}=&{} -\varepsilon X_{\alpha +\beta } - b^2 &{} [D_7,X_{-\alpha }] \!&{}=&{}\! \frac{\varepsilon }{b^2} X_{\alpha +\beta }+H_{1,0}\!-\! H_{0,1},\\ &{} &{} H_{1,0} + b^2H_{0,1},&{}&{}&{}\\ {[}D_7,X_{\beta }] &{}=&{} -X_{\alpha +\beta }, &{} [D_7,X_{-\beta }] \!&{}=&{}\! -b^2 X_{-\alpha -\beta } \!-\! \varepsilon H_{0,1},\\ {[}D_7,X_{\alpha +\beta }] &{}=&{} 2b^2 X_{\beta }- X_{2\alpha +\beta }, &{} [D_7,X_{-\alpha -\beta }] &{}=&{} -\frac{\varepsilon }{b^2} X_{\alpha } + \varepsilon X_{-\alpha }\\ &{}&{}&{}&{}&{} +\, 2X_{-\beta }-b^2 X_{-2\alpha -\beta },\\ {[}D_7,X_{2\alpha +\beta }] &{}=&{} 2b^2 X_{\alpha +\beta }, &{} [D_7,X_{-2\alpha -\beta }] &{}=&{} 2X_{-\alpha -\beta } - \frac{2\varepsilon }{b^2}H_{1,0},\\ {[}D_7,H_{1,0}] &{}=&{} X_{\alpha } - \frac{\varepsilon }{b^2} X_{2\alpha +\beta }-b^2X_{-\alpha }, &{} [D_7,H_{0,1}] \!&{}=&{}\! -X_{\alpha } \!-\! 2\varepsilon X_{\beta }+b^2X_{-\alpha } \end{array} \end{aligned}$$

and

$$\begin{aligned} M_\Gamma = \begin{pmatrix} -\Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\frac{\varepsilon }{b^2} &{}\quad 0 &{}\quad 1 &{}\quad -1 \\ 0 &{}\quad -\Gamma &{}\quad 2b^2 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -2\varepsilon \\ -\varepsilon &{}\quad -1 &{}\quad -\Gamma &{}\quad 2b^2 &{}\quad \frac{\varepsilon }{b^2} &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad -1 &{}\quad -\Gamma &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\frac{\varepsilon }{b^2} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\Gamma &{}\quad 0 &{}\quad \varepsilon &{}\quad 0 &{}\quad -b^2 &{}\quad b^2 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\Gamma &{}\quad 2 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -b^2 &{}\quad -\Gamma &{}\quad 2 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -b^2 &{}\quad -\Gamma &{}\quad 0 &{}\quad 0 \\ -b^2 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad -\frac{2\varepsilon }{b^2} &{}\quad -\Gamma &{}\quad 0 \\ b^2 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -1 &{}\quad -\varepsilon &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\Gamma \end{pmatrix} \end{aligned}$$

with \(\det \, M_\Gamma = \Gamma ^4 (\Gamma ^2 + 4 b^2)^3\), \(b > 0\).

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Dahlke, S., De Mari, F., De Vito, E. et al. Different Faces of the Shearlet Group. J Geom Anal 26, 1693–1729 (2016). https://doi.org/10.1007/s12220-015-9605-7

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