Property of polynomials associated with mirror related representations
Let $mathcal{P}$ be point symmetry group with $Gamma$ and $Gamma'$ being two irreducible representations, that merely differ in the sign of the character of the mirrored sector. Also, let $f_Gamma$ be the associated polynomial to $Gamma$.
As an example consider, the representations $B_1$ and $B_2$ of $mathcal{P}=C_{4v}$, where the lowest order associated polynomials are: $f_{B_1}=x^2-y^2$ and $f_{B_2}=x y$. In polar coordinates, those polynomials become the harmonics $f_{B_1}(phi)=cos2phi$ and $f_{B_2}(phi)=sin2phi$, which satisfy $$f_{B_1}^2+f_{B_2}^2equiv 1.$$
This seems to be a general property since the higher order polynomials of $A_1$ and $A_2$ ($cos4phi$ and $sin4phi$) as well as the one-dimensional irreducible representations of for example $mathcal{P}=C_{6v}$ satisfy this condition.
In one dimension the argumentation seems to rely on the intermediate value theorem in the following way: Functions $f_Gamma$ have to be $0$ at fixed points of mirror operations $sigma$ if the $chi_Gamma(sigma)=-1$ since the function has to be odd around the fixed point. This leads in the examples to a complete decomposition in terms of angular harmonics transforming in the two sectors $Gamma$ and $Gamma'$.
Is here a more rigorous statement possible? Furthermore, can it be generalized to higher dimensions?
finite-groups representation-theory spherical-harmonics
asked Nov 27 '18 at 15:40
user13718
11
add a comment |
Let $mathcal{P}$ be point symmetry group with $Gamma$ and $Gamma'$ being two irreducible representations, that merely differ in the sign of the character of the mirrored sector. Also, let $f_Gamma$ be the associated polynomial to $Gamma$.
As an example consider, the representations $B_1$ and $B_2$ of $mathcal{P}=C_{4v}$, where the lowest order associated polynomials are: $f_{B_1}=x^2-y^2$ and $f_{B_2}=x y$. In polar coordinates, those polynomials become the harmonics $f_{B_1}(phi)=cos2phi$ and $f_{B_2}(phi)=sin2phi$, which satisfy $$f_{B_1}^2+f_{B_2}^2equiv 1.$$
This seems to be a general property since the higher order polynomials of $A_1$ and $A_2$ ($cos4phi$ and $sin4phi$) as well as the one-dimensional irreducible representations of for example $mathcal{P}=C_{6v}$ satisfy this condition.
In one dimension the argumentation seems to rely on the intermediate value theorem in the following way: Functions $f_Gamma$ have to be $0$ at fixed points of mirror operations $sigma$ if the $chi_Gamma(sigma)=-1$ since the function has to be odd around the fixed point. This leads in the examples to a complete decomposition in terms of angular harmonics transforming in the two sectors $Gamma$ and $Gamma'$.
Is here a more rigorous statement possible? Furthermore, can it be generalized to higher dimensions?
finite-groups representation-theory spherical-harmonics
asked Nov 27 '18 at 15:40
user13718
11
add a comment |
Let $mathcal{P}$ be point symmetry group with $Gamma$ and $Gamma'$ being two irreducible representations, that merely differ in the sign of the character of the mirrored sector. Also, let $f_Gamma$ be the associated polynomial to $Gamma$.
As an example consider, the representations $B_1$ and $B_2$ of $mathcal{P}=C_{4v}$, where the lowest order associated polynomials are: $f_{B_1}=x^2-y^2$ and $f_{B_2}=x y$. In polar coordinates, those polynomials become the harmonics $f_{B_1}(phi)=cos2phi$ and $f_{B_2}(phi)=sin2phi$, which satisfy $$f_{B_1}^2+f_{B_2}^2equiv 1.$$
This seems to be a general property since the higher order polynomials of $A_1$ and $A_2$ ($cos4phi$ and $sin4phi$) as well as the one-dimensional irreducible representations of for example $mathcal{P}=C_{6v}$ satisfy this condition.
In one dimension the argumentation seems to rely on the intermediate value theorem in the following way: Functions $f_Gamma$ have to be $0$ at fixed points of mirror operations $sigma$ if the $chi_Gamma(sigma)=-1$ since the function has to be odd around the fixed point. This leads in the examples to a complete decomposition in terms of angular harmonics transforming in the two sectors $Gamma$ and $Gamma'$.
Is here a more rigorous statement possible? Furthermore, can it be generalized to higher dimensions?
finite-groups representation-theory spherical-harmonics
asked Nov 27 '18 at 15:40
user13718
11
Let $mathcal{P}$ be point symmetry group with $Gamma$ and $Gamma'$ being two irreducible representations, that merely differ in the sign of the character of the mirrored sector. Also, let $f_Gamma$ be the associated polynomial to $Gamma$.
As an example consider, the representations $B_1$ and $B_2$ of $mathcal{P}=C_{4v}$, where the lowest order associated polynomials are: $f_{B_1}=x^2-y^2$ and $f_{B_2}=x y$. In polar coordinates, those polynomials become the harmonics $f_{B_1}(phi)=cos2phi$ and $f_{B_2}(phi)=sin2phi$, which satisfy $$f_{B_1}^2+f_{B_2}^2equiv 1.$$
This seems to be a general property since the higher order polynomials of $A_1$ and $A_2$ ($cos4phi$ and $sin4phi$) as well as the one-dimensional irreducible representations of for example $mathcal{P}=C_{6v}$ satisfy this condition.
In one dimension the argumentation seems to rely on the intermediate value theorem in the following way: Functions $f_Gamma$ have to be $0$ at fixed points of mirror operations $sigma$ if the $chi_Gamma(sigma)=-1$ since the function has to be odd around the fixed point. This leads in the examples to a complete decomposition in terms of angular harmonics transforming in the two sectors $Gamma$ and $Gamma'$.
Is here a more rigorous statement possible? Furthermore, can it be generalized to higher dimensions?
finite-groups representation-theory spherical-harmonics
finite-groups representation-theory spherical-harmonics
asked Nov 27 '18 at 15:40
user13718
11
asked Nov 27 '18 at 15:40
user13718
11
asked Nov 27 '18 at 15:40
user13718
11
asked Nov 27 '18 at 15:40
user13718
11
asked Nov 27 '18 at 15:40
user13718
11
11
add a comment |
add a comment |
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