How Much Do You Know About Alvis-curtis Duality?

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Most of the theories that we use today in mathematics are most of the time very old. It is as if all has already been invented or deciphered when it comes to math. Well, Alvis Curtis has proven us wrong because in the 1990s he came up with a new and bright theory. It is described as a duality operation on the characters of a reductive group over a finite field. So, how much do you know about it? Take our quiz and find out.

• 1.

What's duality?

• A.

It's a group of mathematical structures into other concepts.

• B.

It's a group of theorems into other concepts.

• C.

It's a group of operations into other concepts.

• D.

It's a group of concepts, theorems/mathematical structures into other concepts.

D. It's a group of concepts, theorems/mathematical structures into other concepts.
Explanation
Duality refers to a group of concepts, theorems, and mathematical structures that can be transformed or related to other concepts. This suggests that duality involves a connection or correspondence between different mathematical ideas, allowing for a deeper understanding and analysis of the subject matter.

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• 2.

What is a (B, N) pair?

• A.

It is a structure or groups of Lie type that allows 2 to give uniform proofs of many results.

• B.

It is a structure or groups of Lie type that allows 3 to give uniform proofs of many results.

• C.

It is a structure or groups of Lie type that allows 4 to give uniform proofs of many results.

• D.

It is a structure or groups of Lie type that allows 1 to give uniform proofs of many results.

D. It is a structure or groups of Lie type that allows 1 to give uniform proofs of many results.
Explanation
A (B, N) pair refers to a structure or groups of Lie type that allows 1 to give uniform proofs of many results. This means that by utilizing this structure or group, one can provide consistent and generalized proofs for a wide range of outcomes or conclusions.

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• 3.

What's a parabolic induction?

• A.

It's a method of constructing representation of a reductive group from representation of its common subgroups.

• B.

It's a method of constructing representation of a inclusive group from representation of its parabolic subgroups.

• C.

It's a method of constructing representation of a reductive group from representation of its parabolic subgroups.

• D.

It's a method of constructing representation of a reductive group from representation of its subgroups.

C. It's a method of constructing representation of a reductive group from representation of its parabolic subgroups.
Explanation
A parabolic induction is a method of constructing a representation of a reductive group from the representation of its parabolic subgroups. This means that by studying the representations of the parabolic subgroups, we can build a representation of the larger reductive group.

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• 4.

What's the other term for Steinberg representation?

• A.

Steinberg plan

• B.

Steinberg module

• C.

Steinberg multiplication

• D.

B. Steinberg module
Explanation
The other term for Steinberg representation is Steinberg module.

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• 5.

What's the other term for cuspidal representation?

• A.

Cuspidal identity

• B.

Cuspidal formula

• C.

Cuspidal division

• D.

Cuspidal character

D. Cuspidal character
Explanation
The other term for cuspidal representation is cuspidal character. A cuspidal character is a representation of a group that is trivial on all its proper parabolic subgroups. It is a character that cannot be induced from a character of a proper parabolic subgroup.

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• 6.

What's the Gelfand-Graev representation?

• A.

It's a representation of a reductive group over a infinite field.

• B.

It's a representation of a inclusive group over a finite field.

• C.

It's a representation of a reductive group over a finite field.

• D.

It's a representation of a reductive group over a great field.

C. It's a representation of a reductive group over a finite field.
Explanation
The Gelfand-Graev representation is a representation of a reductive group over a finite field. This means that it is a way of expressing the elements of a reductive group (a type of algebraic group) using matrices or linear transformations, where the elements are defined over a finite field. This representation is important in the study of algebraic groups and has applications in various areas of mathematics and theoretical physics.

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• 7.

What is the Alvis Curtis Duality?

• A.

It is an isometry on generalized addition

• B.

It is an isometry on generalized multiplication

• C.

It is an isometry on generalized functions

• D.

It is an isometry on generalized characters

D. It is an isometry on generalized characters
Explanation
The Alvis Curtis Duality refers to an isometry on generalized characters. This means that it preserves the distance between two generalized characters, which are mathematical objects that represent certain properties or features of a system. The Alvis Curtis Duality ensures that the relationship between these generalized characters remains consistent and unchanged, regardless of any transformations or operations applied to them.

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• 8.

What's an idempotent?

• A.

It denotes an element of a set that is unchanged in value when multiplied or otherwise operated on by itself.

• B.

It denotes several elements of a set that is unchanged in value when multiplied or otherwise operated on by itself.

• C.

It denotes 10 elements of a curve that is unchanged in value when added or otherwise operated on by itself.

• D.

It denotes an element of on a curve that is unchanged in value when divided or otherwise operated on by itself.

A. It denotes an element of a set that is unchanged in value when multiplied or otherwise operated on by itself.
Explanation
An idempotent is an element of a set that remains unchanged in value when multiplied or operated on by itself. This means that if you multiply or perform any operation on the element with itself, the result will be the same element.

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• 9.

Who invented the Alvis Curtis duality?

• A.

Charles W.Curtis

• B.

Dean Alvis

• C.

Kawanaka

• D.

Carter

A. Charles W.Curtis
Explanation
Charles W. Curtis is credited with inventing the Alvis Curtis duality.

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• 10.

• A.

1979

• B.

1980

• C.

1950

• D.

1880

B. 1980
Explanation
The theory was invented in 1980.

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• Current Version
• Mar 19, 2023
Quiz Edited by
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• Jul 19, 2018
Quiz Created by
Anouchka

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