In the modular case, Schur algebras were used by which duo to show that the problems of computing decomposition numbers for general linear groups and symmetric groups are actually equivalent?

Michael Redman and Suslin Piers

What Do You Know About Schur Algebras?

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Questions: 10 | Attempts: 117 | Updated: Mar 21, 2023

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What Do You Know About Schur Algebras? - Quiz

Schur algebras are useful in the field of representation theory under the science of structure, order, and relation otherwise known as mathematics.
They are specific limited-dimensional algebras intently connected with Schur–Weyl duality between general symmetric and linear groups.
To find out more about Schur algebras, take this short, intelligent quiz.

Questions and Answers

1.

Schur algebras are named after who?
- A.
  
  Isa Schur
- B.
  
  Issai Schur
- C.
  
  Issac Schur
- D.
  
  Mathew Schur
Correct Answer
B. Issai Schur

Explanation
Schur algebras are named after Issai Schur, a mathematician who made significant contributions to the field of algebra. He introduced Schur algebras in his work on the representation theory of symmetric groups and proved fundamental theorems related to them. His work laid the foundation for the study of Schur algebras and their applications in various areas of mathematics.

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2.

Who widely promoted the use of Schur algebras?
- A.
  
  A. J. Green
- B.
  
  J. A. Green
- C.
  
  C. J. Green
- D.
  
  J. C. Green
Correct Answer
B. J. A. Green

Explanation
J. A. Green widely promoted the use of Schur algebras.

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3.

Schur algebras were used by which duo to prove finite generation of cohomology of finite group schemes?
- A.
  
  Friedlander and Suslin
- B.
  
  Karin and Edman
- C.
  
  Suslin and Edman
- D.
  
  Friedlander and Karin
Correct Answer
A. Friedlander and Suslin

Explanation
Friedlander and Suslin used Schur algebras to prove the finite generation of cohomology of finite group schemes.

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4.

Who put forth generalized Schur algebras?
- A.
  
  Suslin
- B.
  
  Donkin
- C.
  
  Edman
- D.
  
  Karin
Correct Answer
B. Donkin

Explanation
Donkin is the correct answer because he is the mathematician who put forth generalized Schur algebras. Suslin, Edman, and Karin are not associated with this concept.

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5.

In which decade were generalised Schur algebras introduced?
- A.
  
  1970s
- B.
  
  1980s
- C.
  
  1990s
- D.
  
  2000s
Correct Answer
B. 1980s

Explanation
The correct answer is 1980s. Generalized Schur algebras were introduced in the 1980s.

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6.

Which duo introduced quantized Schur algebras?
- A.
  
  Suslin and James
- B.
  
  Edward and Dipper
- C.
  
  James and Karin
- D.
  
  Dipper and James
Correct Answer
D. Dipper and James

Explanation
Dipper and James introduced quantized Schur algebras.

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7.

Schur algebras have which global dimensions?
- A.
  
  Finite
- B.
  
  Infinite
- C.
  
  Semi-finite
- D.
  
  Semi-infinite
Correct Answer
A. Finite

Explanation
Schur algebras have finite global dimensions. This means that the Schur algebras have a finite number of simple modules up to isomorphism. In other words, there are only finitely many irreducible representations of the Schur algebra.

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8.

The name Schur algebra was because of who?
- A.
  
  Suslin
- B.
  
  Issai
- C.
  
  Green
- D.
  
  Karin
Correct Answer
C. Green

Explanation
The correct answer is Green. The name Schur algebra was given in honor of the mathematician Green.

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9.

In which year was the monograph used to introduce Schur algebras published?
- A.
  
  1978
- B.
  
  1979
- C.
  
  1980
- D.
  
  1981
- E.
  
  1982
Correct Answer
C. 1980

Explanation
The monograph used to introduce Schur algebras was published in 1980.

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10.

In the modular case, Schur algebras were used by which duo to show that the problems of computing decomposition numbers for general linear groups and symmetric groups are actually equivalent?
- A.
  
  Gordon James and Karin Erdmann
- B.
  
  James Redman and Suslin Piers
- C.
  
  Gordon James and Karin Redman
- D.
  
  Michael Redman and Suslin Piers
Correct Answer
A. Gordon James and Karin Erdmann

Explanation
Gordon James and Karin Erdmann used Schur algebras to demonstrate that the problems of computing decomposition numbers for general linear groups and symmetric groups are equivalent in the modular case.

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