Which of the following does Schubert polynomials have?

The coefficient depends on the value

What Do You Know About Schubert Polynomials?

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Questions: 10 | Attempts: 129 | Updated: Mar 20, 2023

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What Do You Know About Schubert Polynomials? - Quiz

Schubert polynomials are speculations of Schur polynomials that speak to cohomology classes of Schubert cycles in flag or hail assortments.
In mathematics, there are many variations, which include double Schubert polynomials, quantum Schubert polynomials, and universal Schubert polynomials.
If you'd like to know more about Schubert polynomials, take the short quiz below to assess your knowledge.

Questions and Answers

1.

Schubert polynomials are generalizations of Schur polynomials that represent which of these?
- A.
  
  Trihomology classes
- B.
  
  Dihomology classes
- C.
  
  Homology classes
- D.
  
  Teramology classes
Correct Answer
C. Homology classes

Explanation
Schubert polynomials are generalizations of Schur polynomials, which are polynomials that represent homology classes. Homology classes are fundamental objects in algebraic topology that capture the properties and structure of spaces. Schubert polynomials provide a way to study and compute these homology classes in a more general setting. Therefore, the correct answer is homology classes.

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2.

Who introduced Schubert polynomials?
- A.
  
  Richard and Stanley
- B.
  
  Lascoux and Schutzenberger
- C.
  
  Lascoux and Richard
- D.
  
  Stanley and Hermaine
Correct Answer
B. Lascoux and Schutzenberger

Explanation
Lascoux and Schutzenberger introduced Schubert polynomials.

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3.

After who are Schubert polynomials named?
- A.
  
  Hermaine Schubert
- B.
  
  Hermanne Schubert
- C.
  
  Herman Schubert
- D.
  
  Henry Schubert
Correct Answer
B. Hermanne Schubert

Explanation
Schubert polynomials are named after Hermanne Schubert. The spelling of her name is "Hermanne" and not "Hermaine" or "Herman" or "Henry". Therefore, the correct answer is Hermanne Schubert.

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4.

Which of the following does Schubert polynomials have?
- A.
  
  Negative coefficient
- B.
  
  Positive coefficient
- C.
  
  Neutral coefficient
- D.
  
  The coefficient depends on the value
Correct Answer
B. Positive coefficient

Explanation
Schubert polynomials have positive coefficients. This means that each term in the polynomial has a positive coefficient, indicating that the polynomial is always positive or non-negative.

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5.

Double Schubert polynomials are polynomials in what?
- A.
  
  1 infinite set of variables
- B.
  
  2 infinite sets of variables
- C.
  
  3 infinite sets of variables
- D.
  
  4 infinite set of variables
- E.
  
  No infinite set of variables
Correct Answer
B. 2 infinite sets of variables

Explanation
Double Schubert polynomials are polynomials in two infinite sets of variables. These polynomials are used in algebraic geometry and combinatorics to study the intersection theory on Grassmannians and flag varieties. The two infinite sets of variables correspond to the Schubert cells and the opposite Schubert cells, which are important objects in the study of flag varieties. The double Schubert polynomials encode information about the intersection numbers of these cells and have applications in representation theory and algebraic geometry.

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6.

Which of these individuals put forth a conjectural rule for their coefficients?
- A.
  
  Sara Billey
- B.
  
  Richard P. Stanley
- C.
  
  Sergey Formin
- D.
  
  Lasoux
Correct Answer
B. Richard P. Stanley

Explanation
Richard P. Stanley is the correct answer because he is known for his work in algebraic combinatorics and has made significant contributions to the field. In particular, he has put forth a conjectural rule for the coefficients of certain combinatorial polynomials, which suggests a relationship between various combinatorial objects. This conjecture has been influential in the study of combinatorial structures and has led to further research and developments in the field.

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7.

Who described the history of Schubert polynomials?
- A.
  
  Lascoux
- B.
  
  Jockush
- C.
  
  Sergey
- D.
  
  Hermaine
Correct Answer
A. Lascoux

Explanation
Lascoux is the correct answer because they are the one who described the history of Schubert polynomials.

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8.

In which year was the history of Schubert polynomials described?
- A.
  
  1995
- B.
  
  1996
- C.
  
  1997
- D.
  
  1998
Correct Answer
A. 1995

Explanation
The history of Schubert polynomials was described in the year 1995.

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9.

What do we call the Schubert polynomials that can be seen as a generating function over certain combinatorial objects?
- A.
  
  Pipe dreams or rc-graphs
- B.
  
  Ripe dreams or rd-graphs
- C.
  
  Side dreams or 3d-graphs
- D.
  
  Side dreams or 4s-graphs
Correct Answer
A. Pipe dreams or rc-graphs

Explanation
Pipe dreams or rc-graphs are called the Schubert polynomials that can be seen as a generating function over certain combinatorial objects.

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10.

Fomin, Gelfand, and Postnikov introduced quantum Schubert polynomials in which year?
- A.
  
  1997
- B.
  
  1996
- C.
  
  1995
- D.
  
  1994
Correct Answer
A. 1997

Explanation
Fomin, Gelfand, and Postnikov introduced quantum Schubert polynomials in 1997.

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