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Vít Tuček
Vít Tuček
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I want to find explicit formulae for projectors onto irreducible summands of $\bigotimes^k \mathfrak{g}$, where $\mathfrak{g}$ is a simple Lie algebra. Or more generally, write down the projectors of $\bigotimes^k \mathbb{V}_\lambda$ for irreducible representation of $\mathfrak{g}$ of weight $\lambda$.

What I have found out so far:

A classical result of Weyl on realization of finite-dimensional irreducible representation leads to a nice and quite explicit decomposition of the tensor power $\bigotimes^k \mathbb{C}^n$ of the defining representation of $GL(n,\mathbb{C})$. A key point to the proof is the identification of the centralizer of the action inside $End(\bigotimes^k \mathbb{C}^n)$ (i.e. a commutant subalgebra). This subalgebra is actually a group algebra of the symmetric group $S_k$ which acts by permuting the vectors in the tensor product. Understanding of representation of this algebra helps to understand the multiplicities of of irreducible summands of $\bigotimes^k \mathbb{C}^n$. (A special case of Schur functors I suppose.) All of this can be found in the book by Goodmann and Wallach. There, one can also find a similar treatment for tensor powers of the defining representations of orthogonal and symplectic groups. The commutant subalgebra is then the Brauer algebra and again, the projectors are written down more or less explicitly. By the way, only recently the tensor power of the seven-dimensional representation of $G_2$ was decomposed in a similar manner by Huang and Zhu.

What else is known in this area? Can this approach be generalized to tensor powers of other representations?

Specifically, I want to understand a certain ideal in the universal enveloping algebra of $\mathfrak{su}_n$ and I am trying to find its generators by determining which irreducible summands belongs to the ideal and which not. Accordingly, I am mainly interested in the decomposition of $\bigotimes^k \mathfrak{su}_n$.

I want to find explicit formulae for projectors onto irreducible summands of $\bigotimes^k \mathfrak{g}$, where $\mathfrak{g}$ is a simple Lie algebra. Or more generally, write down the projectors of $\bigotimes^k \mathbb{V}_\lambda$ for irreducible representation of $\mathfrak{g}$ of weight $\lambda$.

What I have found out so far:

A classical result of Weyl on realization of finite-dimensional irreducible representation leads to a nice and quite explicit decomposition of the tensor power $\bigotimes^k \mathbb{C}^n$ of the defining representation of $GL(n,\mathbb{C})$. A key point to the proof is the identification of the centralizer of the action inside $End(\bigotimes^k \mathbb{C}^n)$ (i.e. a commutant subalgebra). This subalgebra is actually a group algebra of the symmetric group $S_k$ which acts by permuting the vectors in the tensor product. Understanding of representation of this algebra helps to understand the multiplicities of of irreducible summands of $\bigotimes^k \mathbb{C}^n$. (A special case of Schur functors I suppose.) All of this can be found in the book by Goodmann and Wallach. There, one can also find a similar treatment for tensor powers of the defining representations of orthogonal and symplectic groups. The commutant subalgebra is then the Brauer algebra and again, the projectors are written down more or less explicitly. By the way, only recently the tensor power of the seven-dimensional representation of $G_2$ was decomposed in a similar manner by Huang and Zhu.

What else is known in this area? Can this approach be generalized to tensor powers of other representations?

I want to find explicit formulae for projectors onto irreducible summands of $\bigotimes^k \mathfrak{g}$, where $\mathfrak{g}$ is a simple Lie algebra. Or more generally, write down the projectors of $\bigotimes^k \mathbb{V}_\lambda$ for irreducible representation of $\mathfrak{g}$ of weight $\lambda$.

What I have found out so far:

A classical result of Weyl on realization of finite-dimensional irreducible representation leads to a nice and quite explicit decomposition of the tensor power $\bigotimes^k \mathbb{C}^n$ of the defining representation of $GL(n,\mathbb{C})$. A key point to the proof is the identification of the centralizer of the action inside $End(\bigotimes^k \mathbb{C}^n)$ (i.e. a commutant subalgebra). This subalgebra is actually a group algebra of the symmetric group $S_k$ which acts by permuting the vectors in the tensor product. Understanding of representation of this algebra helps to understand the multiplicities of of irreducible summands of $\bigotimes^k \mathbb{C}^n$. (A special case of Schur functors I suppose.) All of this can be found in the book by Goodmann and Wallach. There, one can also find a similar treatment for tensor powers of the defining representations of orthogonal and symplectic groups. The commutant subalgebra is then the Brauer algebra and again, the projectors are written down more or less explicitly. By the way, only recently the tensor power of the seven-dimensional representation of $G_2$ was decomposed in a similar manner by Huang and Zhu.

What else is known in this area? Can this approach be generalized to tensor powers of other representations?

Specifically, I want to understand a certain ideal in the universal enveloping algebra of $\mathfrak{su}_n$ and I am trying to find its generators by determining which irreducible summands belongs to the ideal and which not. Accordingly, I am mainly interested in the decomposition of $\bigotimes^k \mathfrak{su}_n$.

(hopefully) clarified the question
Source Link
Vít Tuček
Vít Tuček
  • 8.6k
  • 2
  • 30
  • 61

Schur explicit projectors for tensor powers of general irreducible representation

I want to find explicit formulae for projectors onto irreducible summands of $\bigotimes^k \mathfrak{g}$, where $\mathfrak{g}$ is a simple Lie algebra. Or more generally, write down the projectors of $\bigotimes^k \mathbb{V}_\lambda$ for irreducible representation of $\mathfrak{g}$ of weight $\lambda$.

What I have found out so far:

A classical result of Weyl on realization of finite-dimensional irreducible representation leads to a nice and quite explicit decomposition of the tensor power $\bigotimes^k V$$\bigotimes^k \mathbb{C}^n$ of the defining representation of $G$$GL(n,\mathbb{C})$. TheA key point to the proof proceeds by findingis the Schur dual, which in caseidentification of the centralizer of the action inside $GL(n)$$End(\bigotimes^k \mathbb{C}^n)$ (i.e. a commutant subalgebra). This subalgebra is just theactually a group algebra of the symmetric group $S_k$ andwhich acts by permuting the vectors in the tensor product. Understanding of representation of this algebra helps to understand the multiplicities of of irreducible summands of $\bigotimes^k \mathbb{C}^n$. (A special case of Schur functors I suppose.) All of this can be found in the book by Goodmann and Wallach. There, one can also find a similar treatment for tensor powers of the defining representations of orthogonal and symplectic groupgroups. The commutant subalgebra is then the group algebra of a braid group. One can write down explicit projectors onBrauer algebra and again, the appropriate subrepresentationsprojectors are written down more or less explicitly. This work has been extendedBy the way, only recently to the tensor power of the seven-dimensional representation of $G_2$ was decomposed in a similar manner by Huang and Zhu.

The questionWhat else is whether one has similar result forknown in this area? Can this approach be generalized to tensor powers of any irreducible representation of $G$. I am mainly interested in the projectors onto irreducible summands of $\bigotimes^k \mathfrak{su} (n)$.other representations?

Schur projectors for tensor powers of general irreducible representation

A classical result of Weyl on realization of finite-dimensional irreducible representation leads to a nice and quite explicit decomposition of the tensor power $\bigotimes^k V$ of the defining representation of $G$. The proof proceeds by finding the Schur dual, which in case of $GL(n)$ is just the group algebra of $S_k$ and in case of orthogonal and symplectic group is the group algebra of a braid group. One can write down explicit projectors on the appropriate subrepresentations. This work has been extended only recently to the tensor power of the seven-dimensional representation of $G_2$ by Huang and Zhu.

The question is whether one has similar result for tensor powers of any irreducible representation of $G$. I am mainly interested in the projectors onto irreducible summands of $\bigotimes^k \mathfrak{su} (n)$.

explicit projectors for tensor powers of irreducible representation

I want to find explicit formulae for projectors onto irreducible summands of $\bigotimes^k \mathfrak{g}$, where $\mathfrak{g}$ is a simple Lie algebra. Or more generally, write down the projectors of $\bigotimes^k \mathbb{V}_\lambda$ for irreducible representation of $\mathfrak{g}$ of weight $\lambda$.

What I have found out so far:

A classical result of Weyl on realization of finite-dimensional irreducible representation leads to a nice and quite explicit decomposition of the tensor power $\bigotimes^k \mathbb{C}^n$ of the defining representation of $GL(n,\mathbb{C})$. A key point to the proof is the identification of the centralizer of the action inside $End(\bigotimes^k \mathbb{C}^n)$ (i.e. a commutant subalgebra). This subalgebra is actually a group algebra of the symmetric group $S_k$ which acts by permuting the vectors in the tensor product. Understanding of representation of this algebra helps to understand the multiplicities of of irreducible summands of $\bigotimes^k \mathbb{C}^n$. (A special case of Schur functors I suppose.) All of this can be found in the book by Goodmann and Wallach. There, one can also find a similar treatment for tensor powers of the defining representations of orthogonal and symplectic groups. The commutant subalgebra is then the Brauer algebra and again, the projectors are written down more or less explicitly. By the way, only recently the tensor power of the seven-dimensional representation of $G_2$ was decomposed in a similar manner by Huang and Zhu.

What else is known in this area? Can this approach be generalized to tensor powers of other representations?

Source Link
Vít Tuček
Vít Tuček
  • 8.6k
  • 2
  • 30
  • 61

Schur projectors for tensor powers of general irreducible representation

A classical result of Weyl on realization of finite-dimensional irreducible representation leads to a nice and quite explicit decomposition of the tensor power $\bigotimes^k V$ of the defining representation of $G$. The proof proceeds by finding the Schur dual, which in case of $GL(n)$ is just the group algebra of $S_k$ and in case of orthogonal and symplectic group is the group algebra of a braid group. One can write down explicit projectors on the appropriate subrepresentations. This work has been extended only recently to the tensor power of the seven-dimensional representation of $G_2$ by Huang and Zhu.

The question is whether one has similar result for tensor powers of any irreducible representation of $G$. I am mainly interested in the projectors onto irreducible summands of $\bigotimes^k \mathfrak{su} (n)$.