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| − | The (left) regular representation of an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r0808101.png" /> is the [[Linear representation|linear representation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r0808102.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r0808103.png" /> on the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r0808104.png" /> defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r0808105.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r0808106.png" />. Similarly, the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r0808107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r0808108.png" />, defines an (anti-) representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r0808109.png" /> on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081010.png" />, called the (right) regular representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081011.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081012.png" /> is a topological algebra (with continuous multiplication in all the variables), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081014.png" /> are continuous representations. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081015.png" /> is an algebra with a unit element or a semi-simple algebra, then its regular representations are faithful (cf. [[Faithful representation|Faithful representation]]). | + | {{TEX|done}} |
| | + | The (left) regular representation of an algebra $A$ is the [[Linear representation|linear representation]] $L$ of $A$ on the vector space $E=A$ defined by the formula $L(a)b=ab$ for all $a,b\in A$. Similarly, the formula $R(a)b=ba$, $a,b\in a$, defines an (anti-) representation of $A$ on the space $E=A$, called the (right) regular representation of $A$. If $A$ is a topological algebra (with continuous multiplication in all the variables), then $L$ and $R$ are continuous representations. If $A$ is an algebra with a unit element or a semi-simple algebra, then its regular representations are faithful (cf. [[Faithful representation|Faithful representation]]). |
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| − | A (right) regular representation of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081016.png" /> is a linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081018.png" /> on a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081019.png" /> of complex-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081020.png" />, defined by the formula | + | A (right) regular representation of a group $G$ is a linear representation $R$ of $G$ on a space $E$ of complex-valued functions on $G$, defined by the formula |
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081021.png" /></td> </tr></table>
| + | $$(R(g)f)(g_1)=f(g_1g),\quad g,g_1\in G,\quad f\in E,$$ |
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| − | provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081022.png" /> separates the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081023.png" /> and has the property that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081025.png" />, belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081026.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081028.png" />. Similarly, the formula | + | provided that $E$ separates the points of $G$ and has the property that the function $g_1\mapsto f(g_1g)$, $g_1\in G$, belongs to $E$ for all $f\in E$, $g\in G$. Similarly, the formula |
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081029.png" /></td> </tr></table>
| + | $$(L(g)f)(g_1)=f(g^{-1}g_1),\quad g,g_1\in G,\quad f\in E,$$ |
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| − | defines a (left) regular representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081030.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081031.png" />, where the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081033.png" />, is assumed to belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081034.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081036.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081037.png" /> is a topological group, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081038.png" /> is often the space of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081039.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081040.png" /> is locally compact, then the (right) regular representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081041.png" /> is the (right) regular representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081042.png" /> on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081043.png" /> constructed by means of the right-invariant [[Haar measure|Haar measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080810/r08081044.png" />; the regular representation of a locally compact group is a continuous [[Unitary representation|unitary representation]], and the left and right regular representations are unitarily equivalent. | + | defines a (left) regular representation of $G$ on $E$, where the function $g\mapsto f(g^{-1}g_1)$, $g_1\in G$, is assumed to belong to $E$ for all $g\in G$, $f\in E$. If $G$ is a topological group, then $E$ is often the space of continuous functions on $G$. If $G$ is locally compact, then the (right) regular representation of $G$ is the (right) regular representation of $G$ on the space $L_2(G)$ constructed by means of the right-invariant [[Haar measure|Haar measure]] on $G$; the regular representation of a locally compact group is a continuous [[Unitary representation|unitary representation]], and the left and right regular representations are unitarily equivalent. |
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| | ====Comments==== | | ====Comments==== |
| − | | + | For a finite group $G$, the action of $G$ on the [[group ring]] $\mathbb{C}G$ gives the regular representation of $G$. This representation contains a copy of each of the irreducible representations of $G$. |
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| | ====References==== | | ====References==== |
| | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C.W. Curtis, I. Reiner, "Methods of representation theory" , '''1–2''' , Wiley (Interscience) (1981–1987)</TD></TR></table> | | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C.W. Curtis, I. Reiner, "Methods of representation theory" , '''1–2''' , Wiley (Interscience) (1981–1987)</TD></TR></table> |
The (left) regular representation of an algebra $A$ is the linear representation $L$ of $A$ on the vector space $E=A$ defined by the formula $L(a)b=ab$ for all $a,b\in A$. Similarly, the formula $R(a)b=ba$, $a,b\in a$, defines an (anti-) representation of $A$ on the space $E=A$, called the (right) regular representation of $A$. If $A$ is a topological algebra (with continuous multiplication in all the variables), then $L$ and $R$ are continuous representations. If $A$ is an algebra with a unit element or a semi-simple algebra, then its regular representations are faithful (cf. Faithful representation).
A (right) regular representation of a group $G$ is a linear representation $R$ of $G$ on a space $E$ of complex-valued functions on $G$, defined by the formula
$$(R(g)f)(g_1)=f(g_1g),\quad g,g_1\in G,\quad f\in E,$$
provided that $E$ separates the points of $G$ and has the property that the function $g_1\mapsto f(g_1g)$, $g_1\in G$, belongs to $E$ for all $f\in E$, $g\in G$. Similarly, the formula
$$(L(g)f)(g_1)=f(g^{-1}g_1),\quad g,g_1\in G,\quad f\in E,$$
defines a (left) regular representation of $G$ on $E$, where the function $g\mapsto f(g^{-1}g_1)$, $g_1\in G$, is assumed to belong to $E$ for all $g\in G$, $f\in E$. If $G$ is a topological group, then $E$ is often the space of continuous functions on $G$. If $G$ is locally compact, then the (right) regular representation of $G$ is the (right) regular representation of $G$ on the space $L_2(G)$ constructed by means of the right-invariant Haar measure on $G$; the regular representation of a locally compact group is a continuous unitary representation, and the left and right regular representations are unitarily equivalent.
For a finite group $G$, the action of $G$ on the group ring $\mathbb{C}G$ gives the regular representation of $G$. This representation contains a copy of each of the irreducible representations of $G$.
References
| [a1] | C.W. Curtis, I. Reiner, "Methods of representation theory" , 1–2 , Wiley (Interscience) (1981–1987) |