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Template:Ns$\underset{a}{\overset{b}{\int }}f(x)dx$

The Template:Integrate can format a math-tag integral for parameter 1, with optional parameters "from=" or "to=" or "dx=".

Format of math-tag:Template:Ns<math>\int_a^b \! f(x)\,dx \,</math>

The default is an integral for f(x), from a to b. The amount of indentation can be reset by "indent=0" (a count of spaces). Examples:

• {{integrate|g(x) }} → Template:Multiple issues

Definition

An isotypical or primary representation of a group G is a unitary representation $\pi :G⟶\mathcal{ℬ}(\mathcal{\mathscr{H}})$ such that any two subrepresentations have equivalent subsubrepresentations.

This is to relate to primary or factor representation of a C*-algebra, or to the notion of factor for a von Neumann algebra: the representation $\pi$ of G is isotypicall iff $\pi (G{)}^{{\text{'}}^{\prime }}$ is a factor.

This term more generally used in the context of semisimple module.

Example

Let G be a compact group. A corollary of the Peter-Weyl theorem has that any unitary representation $\pi :G⟶\mathcal{ℬ}(\mathcal{\mathscr{H}})$ on a separable Hilbert space $\mathcal{\mathscr{H}}$ is a possibly infinite direct sum of finite dimensional irreducible representations. An isotypical representation is a direct sum of the equivalent irreducible representations that appear, possibly multiple times, in $\mathcal{\mathscr{H}}$.

References

Mackey

"C* algebras", Jacques Dixmier, Chapter 5

"Lie Groups", Claudio Procesi, def. p. 156.

Template:Abstract-algebra-stub

• {{integrate|f(x,y)|dx=dx\,dy }} → Template:Multiple issues

Definition

An isotypical or primary representation of a group G is a unitary representation $\pi :G⟶\mathcal{ℬ}(\mathcal{\mathscr{H}})$ such that any two subrepresentations have equivalent subsubrepresentations.

This is to relate to primary or factor representation of a C*-algebra, or to the notion of factor for a von Neumann algebra: the representation $\pi$ of G is isotypicall iff $\pi (G{)}^{{\text{'}}^{\prime }}$ is a factor.

This term more generally used in the context of semisimple module.

Example

Let G be a compact group. A corollary of the Peter-Weyl theorem has that any unitary representation $\pi :G⟶\mathcal{ℬ}(\mathcal{\mathscr{H}})$ on a separable Hilbert space $\mathcal{\mathscr{H}}$ is a possibly infinite direct sum of finite dimensional irreducible representations. An isotypical representation is a direct sum of the equivalent irreducible representations that appear, possibly multiple times, in $\mathcal{\mathscr{H}}$.

References

Mackey

"C* algebras", Jacques Dixmier, Chapter 5

"Lie Groups", Claudio Procesi, def. p. 156.

Template:Abstract-algebra-stub

When displaying the integral, there might be a delay as the math-tag is being formatted into the requested symbols. A re-display of a prior, common formula will be slightly faster than showing a new formula.

This template demonstrates the use of the #tag-function to allow generation of a math-tag based on various parameters passed into a template.

See also

• Help:Displaying a formula - explains dozens of options used in math-tags