Schur functor - Revision history

Vipul: /* Facts */

2012-06-17T05:23:53Z

Facts

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	where the latter <math>\mathbb{S}_\rho</math> denotes the effect of the functor on morphisms.		where the latter <math>\mathbb{S}_\rho</math> denotes the effect of the functor on morphisms.

			==Relation with Schur elements and Schur polynomials==

			For an unordered integer partition <math>\lambda</math> of <math>n</math> and an unrelated positive integer <matH>m</math>, the following is the relation between the Schur functor, Schur element, and Schur polynomial:

			* The Schur ''functor'' gives the vector space with basis the semistandard Young tableaux.
			* The Schur ''element'' is the sum of the semistandard Young tableaux.
			* The Schur ''polynomial'' is the image of the Schur element under the mapping to the polynomial ring that only remembers the multiplicity of each variable.

			In particular, the ''dimension'' of the vector space obtained by applying the Schur functor can be obtained from the Schur polynomial by plugging in the value 1 for all the variables.

	==Facts==		==Facts==

Vipul: /* Definition as a functor from vector spaces to vector spaces */

2012-06-17T05:20:18Z

Definition as a functor from vector spaces to vector spaces

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	The result also holds if the characteristic of <math>K</math> is greater than <math>n</math>. If <math>K</math> has characteristic a prime less than or equal to <math>n</math>, there is some ambiguity about how to define the Schur functor.		The result also holds if the characteristic of <math>K</math> is greater than <math>n</math>. If <math>K</math> has characteristic a prime less than or equal to <math>n</math>, there is some ambiguity about how to define the Schur functor.

			===Combinatorial definition for finite-dimensional vector space in terms of Young tableaux===

			The Schur functor corresponding to a partition <math>\lambda</math> of <math>n</math> applied to the vector space <math>K^m</math> (with <math>m</math> unrelated to <math>n</math>) can be defined combinatorially as follows. As a vector space, it has basis the set of semistandard Young tableaux of shape <math>\lambda</math> and all entries between 1 and <math>m</math>.

	==Definition as a functor from representations to representations==		==Definition as a functor from representations to representations==

Vipul: /* Facts */

2012-06-17T04:36:51Z

Facts

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	* We have the following:		* We have the following:

	<math>V^{\otimes n} = \bigoplus_~~\lambda d_~~\lambda \mathbb{S}_\lambda</math>		<math>V^{\otimes n} = \bigoplus_\lambda \mathbb{S}_\lambda^{d_\lambda}</math>

	where <math>d_\lambda</math> denotes the degree of the irreducible representation of <math>S_n</math> corresponding to <math>\lambda</math>.		where <math>d_\lambda</math> denotes the degree of the irreducible representation of <math>S_n</math> corresponding to <math>\lambda</math>. The notation here means that <math>\mathbb{S}_\lambda</math> is added to itself <math>d_\lambda</math> times.

			Note that although <math>\mathbb{S}_\lambda</math> does not have a natural <math>S_n</math>-action, <math>\mathbb{S}_\lambda^{d_\lambda}</math> does.

Vipul: /* Facts */

2012-06-17T04:34:13Z

Facts

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	* [[Formula for calculating effect of Schur functor on character]]		* [[Formula for calculating effect of Schur functor on character]]
			* We have the following:

			<math>V^{\otimes n} = \bigoplus_\lambda d_\lambda \mathbb{S}_\lambda</math>

			where <math>d_\lambda</math> denotes the degree of the irreducible representation of <math>S_n</math> corresponding to <math>\lambda</math>.

Vipul: /* Schur functor associated with a representation of the symmetric group */

2012-06-17T02:27:42Z

Schur functor associated with a representation of the symmetric group

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	===Schur functor associated with a representation of the symmetric group===		===Schur functor associated with a representation of the symmetric group===

	Suppose <math>\rho:S_n \to GL(W)</math> is a [[linear representation]] of the [[symmetric group]] <math>S_n</math> over a [[field]] <math>K</math>. The Schur functor associated with <math>\rho</math> is a functor from the category of <math>K</math>-vector spaces to the category of <math>K</math>-vector spaces defined as follows. First, using the [[equivalence of definitions of linear representation]], consider <math>W</math> as a module over the [[group ring]] <math>KS_n</math>. Now, the functor is defined as follows:		Suppose <math>\rho:S_n \to GL(W)</math> is a [[linear representation]] of the [[symmetric group]] <math>S_n</math> over a [[field]] <math>K</math>. The Schur functor associated with <math>\rho</math>, denoted <math>\mathbb{S}_\rho</math>, is a functor from the category of <math>K</math>-vector spaces to the category of <math>K</math>-vector spaces defined as follows. First, using the [[equivalence of definitions of linear representation]], consider <math>W</math> as a module over the [[group ring]] <math>KS_n</math>. Now, the functor is defined as follows:

	* On objects: It sends a vector space <math>V</math> to the [[tensor product of modules]] <math>V^{\otimes n} \otimes_{KS_n} W</math>. Here is what this means. <math>V^{\otimes n}</math> is the <math>n</math>-fold tensor product of <math>V</math> as a <math>K</math>-vector space. This naturally acquires the structure of a <math>KS_n</math>-module where <math>S_n</math> acts by permuting the tensor product factors. We now take its tensor product ''as a <math>KS_n</math>-module'' with <math>W</math>. This is not the same as taking the tensor product of vector spaces. Note that since <math>KS_n</math> is not commutative, we have to treat <math>W</math> as a left <math>KS_n</math>-module and <math>V^{\otimes n}</math> as a right <math>KS_n</math>-module. Also note that once we have taken the tensor product, we no longer have a <math>KS_n</math>-module structure, just a <math>K</math>-vector space.		* On objects: It sends a vector space <math>V</math> to the [[tensor product of modules]] <math>V^{\otimes n} \otimes_{KS_n} W</math>. Here is what this means. <math>V^{\otimes n}</math> is the <math>n</math>-fold tensor product of <math>V</math> as a <math>K</math>-vector space. This naturally acquires the structure of a <math>KS_n</math>-module where <math>S_n</math> acts by permuting the tensor product factors. We now take its tensor product ''as a <math>KS_n</math>-module'' with <math>W</math>. This is not the same as taking the tensor product of vector spaces. Note that since <math>KS_n</math> is not commutative, we have to treat <math>W</math> as a left <math>KS_n</math>-module and <math>V^{\otimes n}</math> as a right <math>KS_n</math>-module. Also note that once we have taken the tensor product, we no longer have a <math>KS_n</math>-module structure, just a <math>K</math>-vector space.

Vipul: /* Definition as a functor from representations to representations */

2012-06-17T02:27:01Z

Definition as a functor from representations to representations

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	==Definition as a functor from representations to representations==		==Definition as a functor from representations to representations==

	This definition can be obtained using abstract nonsense from the definition for vector spaces. {{~~fillin~~}}		This definition can be obtained using abstract nonsense from the definition for vector spaces. For a representation <math>\alpha:G \to GL(V)</math> and a linear representation <matH>\rho</math> of a symmetric group <math>S_n</math>, we define <math>\mathbb{S}_\rho(\alpha)</math> as a representation of <math>G</math> on the vector space <math>S_\rho(V)</math>, with the map:

			<math>G \to \mathbb{S}_\rho(V)</math>

			given by:

			<math>g \mapsto \mathbb{S}_\rho(\alpha(g))</math>

			where the latter <math>\mathbb{S}_\rho</math> denotes the effect of the functor on morphisms.

	==Facts==		==Facts==

	* [[Formula for calculating effect of Schur functor on character]]		* [[Formula for calculating effect of Schur functor on character]]

Vipul: /* Definition as a functor from representations to reprsentations */

2012-06-17T02:21:09Z

Definition as a functor from representations to reprsentations

← Older revision		Revision as of 02:21, 17 June 2012
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	The result also holds if the characteristic of <math>K</math> is greater than <math>n</math>. If <math>K</math> has characteristic a prime less than or equal to <math>n</math>, there is some ambiguity about how to define the Schur functor.		The result also holds if the characteristic of <math>K</math> is greater than <math>n</math>. If <math>K</math> has characteristic a prime less than or equal to <math>n</math>, there is some ambiguity about how to define the Schur functor.

	==Definition as a functor from representations to ~~reprsentations~~==		==Definition as a functor from representations to representations==

	This definition can be obtained using abstract nonsense from the definition for vector spaces. {{fillin}}		This definition can be obtained using abstract nonsense from the definition for vector spaces. {{fillin}}

Vipul at 02:14, 17 June 2012

2012-06-17T02:14:45Z

← Older revision		Revision as of 02:14, 17 June 2012
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	The result also holds if the characteristic of <math>K</math> is greater than <math>n</math>. If <math>K</math> has characteristic a prime less than or equal to <math>n</math>, there is some ambiguity about how to define the Schur functor.		The result also holds if the characteristic of <math>K</math> is greater than <math>n</math>. If <math>K</math> has characteristic a prime less than or equal to <math>n</math>, there is some ambiguity about how to define the Schur functor.

			==Definition as a functor from representations to reprsentations==

			This definition can be obtained using abstract nonsense from the definition for vector spaces. {{fillin}}

			==Facts==

			* [[Formula for calculating effect of Schur functor on character]]

Vipul: Created page with "==Definition as a functor from vector spaces to vector spaces== ===Schur functor associated with a representation of the symmetric group=== Suppose $\rho:S_n \to GL(W)<..."$

2012-06-17T02:07:20Z

Created page with "==Definition as a functor from vector spaces to vector spaces== ===Schur functor associated with a representation of the symmetric group=== Suppose <math>\rho:S_n \to GL(W)<..."

New page

==Definition as a functor from vector spaces to vector spaces==

===Schur functor associated with a representation of the symmetric group===

Suppose <math>\rho:S_n \to GL(W)</math> is a [[linear representation]] of the [[symmetric group]] <math>S_n</math> over a [[field]] <math>K</math>. The Schur functor associated with <math>\rho</math> is a functor from the category of <math>K</math>-vector spaces to the category of <math>K</math>-vector spaces defined as follows. First, using the [[equivalence of definitions of linear representation]], consider <math>W</math> as a module over the [[group ring]] <math>KS_n</math>. Now, the functor is defined as follows:

* On objects: It sends a vector space <math>V</math> to the [[tensor product of modules]] <math>V^{\otimes n} \otimes_{KS_n} W</math>. Here is what this means. <math>V^{\otimes n}</math> is the <math>n</math>-fold tensor product of <math>V</math> as a <math>K</math>-vector space. This naturally acquires the structure of a <math>KS_n</math>-module where <math>S_n</math> acts by permuting the tensor product factors. We now take its tensor product ''as a <math>KS_n</math>-module'' with <math>W</math>. This is not the same as taking the tensor product of vector spaces. Note that since <math>KS_n</math> is not commutative, we have to treat <math>W</math> as a left <math>KS_n</math>-module and <math>V^{\otimes n}</math> as a right <math>KS_n</math>-module. Also note that once we have taken the tensor product, we no longer have a <math>KS_n</math>-module structure, just a <math>K</math>-vector space.
* On morphisms: Given a linear map <math>\mu:V_1 \to V_2</math>, it induces a map <math>\mu^{\otimes n}:V_1^{\otimes n} \to V_2^{\otimes n}</math> which in turn induces a map of the tensor products with <math>W</math>. Functoriality must be checked, but is true.

===Schur functor associated with a partition===

Assume that <math>K</math> has characteristic zero. Suppose <math>\lambda</math> is a [[set of unordered integer partitions|unordered integer partition]] of a positive integer <math>n</math>. The Schur functor associated with <math>\lambda</math>, denoted <math>\mathbb{S}_\lambda</math>, is defined as the Schur functor associated with the [[irreducible linear representation]] of <math>S_n</math> corresponding to <math>\lambda</math> (see [[linear representation theory of symmetric groups]]).

In this case, the module for the irreducible representation can be thought of explicitly as a left ideal inside <math>KS_n</math> and the tensor product operation can be thought of as multiplication on the right by this ideal. Note that multiplication by the two-sided ideal corresponding to the representation would give a sum of the Schur functor with itslf degree of the representation many times.

The result also holds if the characteristic of <math>K</math> is greater than <math>n</math>. If <math>K</math> has characteristic a prime less than or equal to <math>n</math>, there is some ambiguity about how to define the Schur functor.

Schur functor - Revision history

Vipul: /* Facts */

Vipul: /* Definition as a functor from vector spaces to vector spaces */

Vipul: /* Facts */

Vipul: /* Facts */

Vipul: /* Schur functor associated with a representation of the symmetric group */

Vipul: /* Definition as a functor from representations to representations */

Vipul: /* Definition as a functor from representations to reprsentations */

Vipul at 02:14, 17 June 2012

Vipul: Created page with "==Definition as a functor from vector spaces to vector spaces== ===Schur functor associated with a representation of the symmetric group=== Suppose \rho:S_n \to GL(W)<..."

Vipul: Created page with "==Definition as a functor from vector spaces to vector spaces== ===Schur functor associated with a representation of the symmetric group=== Suppose $\rho:S_n \to GL(W)<..."$