CoCalc -- Lab6-Symmetric-Functions.sagews

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Math 737 - Lab 6

In this lab, we will study the algebra of symmetric functions Sym and its various bases. Evaluate the block below to define the commands for Sym and the bases we discussed in class: $m,e,h,s,p$ .

Sym = SymmetricFunctions(QQ)
s = Sym.schur()
e = Sym.elementary()
h = Sym.homogeneous()
m = Sym.monomial()
p = Sym.powersum()

s(e([2,1]))

s[1, 1, 1] + s[2, 1]

Exercise 1:

Express the monomial symmetric function $m(\lambda)$ for $\lambda=322$ in the basis of Schur functions using the command s(m[3,2,2]).

Do the same for some example elementary, homogeneous, and power sum symmetric functions (at least two of each kind).

Which bases do you think are Schur-positive? (This means the coefficients you get when expressing them in the Schur basis are always positive.)

s(m[3,2,2])

3*s[1, 1, 1, 1, 1, 1, 1] - 3*s[2, 1, 1, 1, 1, 1] + 2*s[2, 2, 1, 1, 1] - s[2, 2, 2, 1] + s[3, 1, 1, 1, 1] - s[3, 2, 1, 1] + s[3, 2, 2]

s(m[2,1])

-2*s[1, 1, 1] + s[2, 1]

Exercise 2: Verify the identities of Exercise 1 on p.74 for values of $n$ up to at least 5.

2*e([2])-p([1])*e([1])+p([2])

0

Exercise 3: Verify Equation $(5)$ on p.75 for $\lambda=[2,1]$ and another larger partition of your choice.

Exercise 4: Sage has a command for the inner product $\langle f,g \rangle$ : f.scalar(g).

Compute some examples of this product on various symmetric functions. Make sure you find some examples for which the inner product is not $0$ or $1$ .
Compute an example that demonstrates the symmetry: $\langle f,g \rangle= \langle g,f \rangle$
Compute examples demonstrating that its action on the Schur basis matches the definition given in the video lecture and the book.
Compute examples demonstrating properties 3. and 4. from the Theorem in the video lecture (about the action on the $p,m,$ and $h$ bases).

Exercise 5: Sage has a command for the $\omega$ involution: .omega().

Compute some examples of the properties discussed in the video lecture of the action of the omega involution on the $s,h,$ and $e$ bases.
Demonstrate that $\omega$ is an involution.
Demonstrate that $\omega$ is an isometry (defined at the end of the video lecture).


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CoCalc provides the best real-time collaborative environment for Jupyter Notebooks, LaTeX documents, and SageMath, scalable from individual use to large groups and classes! Also, H100 GPUs starting at $2/hour.

Symmetric functions

Math 737 - Lab 6

In this lab, we will study the algebra of symmetric functions Sym and its various bases. Evaluate the block below to define the commands for Sym and the bases we discussed in class: $m,e,h,s,p$ .

Exercise 1:

Express the monomial symmetric function $m(\lambda)$ for $\lambda=322$ in the basis of Schur functions using the command s(m[3,2,2]).

Do the same for some example elementary, homogeneous, and power sum symmetric functions (at least two of each kind).

Which bases do you think are Schur-positive? (This means the coefficients you get when expressing them in the Schur basis are always positive.)

Exercise 2: Verify the identities of Exercise 1 on p.74 for values of $n$ up to at least 5.

Exercise 3: Verify Equation $(5)$ on p.75 for $\lambda=[2,1]$ and another larger partition of your choice.

Exercise 4: Sage has a command for the inner product $\langle f,g \rangle$ : f.scalar(g).

Compute some examples of this product on various symmetric functions. Make sure you find some examples for which the inner product is not $0$ or $1$ .

Compute an example that demonstrates the symmetry: $\langle f,g \rangle= \langle g,f \rangle$

Compute examples demonstrating that its action on the Schur basis matches the definition given in the video lecture and the book.

Compute examples demonstrating properties 3. and 4. from the Theorem in the video lecture (about the action on the $p,m,$ and $h$ bases).

Exercise 5: Sage has a command for the $\omega$ involution: .omega().

Compute some examples of the properties discussed in the video lecture of the action of the omega involution on the $s,h,$ and $e$ bases.

Demonstrate that $\omega$ is an involution.

Demonstrate that $\omega$ is an isometry (defined at the end of the video lecture).

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CoCalc provides the best real-time collaborative environment for Jupyter Notebooks, LaTeX documents, and SageMath, scalable from individual use to large groups and classes! Also, H100 GPUs starting at $2/hour.

Symmetric functions

Math 737 - Lab 6

In this lab, we will study the algebra of symmetric functions Sym and its various bases. Evaluate the block below to define the commands for Sym and the bases we discussed in class: m,e,h,s,pm,e,h,s,pm,e,h,s,p.

Exercise 1:

Express the monomial symmetric function m(λ)m(\lambda)m(λ) for λ=322\lambda=322λ=322 in the basis of Schur functions using the command s(m[3,2,2]).

Do the same for some example elementary, homogeneous, and power sum symmetric functions (at least two of each kind).

Which bases do you think are Schur-positive? (This means the coefficients you get when expressing them in the Schur basis are always positive.)

Exercise 2: Verify the identities of Exercise 1 on p.74 for values of nnn up to at least 5.

Exercise 3: Verify Equation (5)(5)(5) on p.75 for λ=[2,1]\lambda=[2,1]λ=[2,1] and another larger partition of your choice.

Exercise 4: Sage has a command for the inner product ⟨f,g⟩\langle f,g \rangle⟨f,g⟩: f.scalar(g).

Compute some examples of this product on various symmetric functions. Make sure you find some examples for which the inner product is not 000 or 111.

Compute an example that demonstrates the symmetry: ⟨f,g⟩=⟨g,f⟩\langle f,g \rangle= \langle g,f \rangle⟨f,g⟩=⟨g,f⟩

Compute examples demonstrating that its action on the Schur basis matches the definition given in the video lecture and the book.

Compute examples demonstrating properties 3. and 4. from the Theorem in the video lecture (about the action on the p,m,p,m,p,m, and hhh bases).

Exercise 5: Sage has a command for the ω\omegaω involution: .omega().

Compute some examples of the properties discussed in the video lecture of the action of the omega involution on the s,h,s,h,s,h, and eee bases.

Demonstrate that ω\omegaω is an involution.

Demonstrate that ω\omegaω is an isometry (defined at the end of the video lecture).

In this lab, we will study the algebra of symmetric functions Sym and its various bases. Evaluate the block below to define the commands for Sym and the bases we discussed in class: $m,e,h,s,p$ .

Express the monomial symmetric function $m(\lambda)$ for $\lambda=322$ in the basis of Schur functions using the command s(m[3,2,2]).

Exercise 2: Verify the identities of Exercise 1 on p.74 for values of $n$ up to at least 5.

Exercise 3: Verify Equation $(5)$ on p.75 for $\lambda=[2,1]$ and another larger partition of your choice.

Exercise 4: Sage has a command for the inner product $\langle f,g \rangle$ : f.scalar(g).

Compute some examples of this product on various symmetric functions. Make sure you find some examples for which the inner product is not $0$ or $1$ .

Compute an example that demonstrates the symmetry: $\langle f,g \rangle= \langle g,f \rangle$

Compute examples demonstrating properties 3. and 4. from the Theorem in the video lecture (about the action on the $p,m,$ and $h$ bases).

Exercise 5: Sage has a command for the $\omega$ involution: .omega().

Compute some examples of the properties discussed in the video lecture of the action of the omega involution on the $s,h,$ and $e$ bases.

Demonstrate that $\omega$ is an involution.

Demonstrate that $\omega$ is an isometry (defined at the end of the video lecture).