Math Problems Vault

Problem 4.101

Let $P = \left\{ P_0, P_2, \ldots , P_{k-1} \right\}$ be a semi-group of permutations on the set $\Omega = \left\{ 0,1,\ldots ,n-1 \right\}$ . A semi-group is closed under the group operation, but the inverse and identity need not be a member of the set. The transitivity set (or orbit) containing $i \in \Omega$ is the set of images under the action of products of elements of $P$ . Write a C++ program which finds the orbits given $P$ and $\Omega$ .

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Problem 4.102

Let $\mathcal{B}_n$ denote the braid group on $n-1$ strands. $\mathcal{B}_n$ is generated by the elementary braids (generators) $\left\{ b_1, b_2, \ldots , b_{n-1} \right\}$ with the braid relations

b_j b_{j+1} b_j = b_{j+1} b_j b_{j+1}, \quad 1 \leq j < n-1,

b_j b_k = b_k b_j, \quad \left|j-k\right| \geq 2 .

Actually one should better write $b_{12}, b_{23}, \ldots , b_{n-1n}$ instead of $b_1, b_2, \ldots , b_{n-1}$ . Let $\left\{ \overrightarrow {e}_1, \overrightarrow {e}_2, \ldots , \overrightarrow {e}_n \right\}$ denote the standard basis in $\mathbb {R}^{n}$ . Then $\overrightarrow {u} \in \mathbb {R}^{n}$ can be written as

\overrightarrow {u} = \sum _{k=1}^{n} c_k \overrightarrow {e}_k, \quad c_1, c_2, \ldots , c_n \in \mathbb {R} .

Consider the operators $B_j$ ( $\alpha ,\beta ,\gamma ,\delta \in \mathbb {R}$ and $\alpha ,\gamma \neq 0$ ) defined by

B_j \overrightarrow {u} := c_1\overrightarrow {e}_1 + \ldots + (\alpha c_{j+1}+\beta )\overrightarrow {e}_j + (\gamma c_{j+1}+\delta )\overrightarrow {e}_{j+1} + \ldots + c_n\overrightarrow {e}_n

and the corresponding inverse operation

B_j^{-1} \overrightarrow {u} := c_1\overrightarrow {e}_1 + \ldots + \frac{1}{\gamma }(c_{j+1}-\delta )\overrightarrow {e}_j + \frac{1}{\alpha }(\gamma c_j-\beta )\overrightarrow {e}_{j+1} + \ldots + c_n\overrightarrow {e}_n .

Use computer algebra to show that $B_1, B_2, \ldots , B_{n-1}$ satisfy the braid condition

B_j B_{j+1} B_j \overrightarrow {u} = B_{j+1} B_j B_{j+1} \overrightarrow {u}

if

\gamma \beta + \delta = \alpha \delta + \beta .

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Problem 4.103

Write a C++ program that generates all $n \times n$ permutation matrices. In main we use these matrices to find the permutation matrix that satisfies

P \begin{pmatrix} a_{00} & 0 & 0 & a_{01} \\ 0 & b_{00} & b_{01} & 0 \\ 0 & b_{10} & b_{11} & 0 \\ a_{10} & 0 & 0 & a_{11} \end{pmatrix} P^T = \begin{pmatrix} a_{00} & a_{01} \\ a_{10} & a_{11} \end{pmatrix} \oplus \begin{pmatrix} b_{00} & b_{01} \\ b_{10} & b_{11} \end{pmatrix}

where $\oplus$ is the direct sum.

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Problem 4.104

(i)(ii)

Let $n$ be a positive integer. There are exactly as many irreducible representations of the permutation group $S_n$ as there are partitions $\lambda_j$ of $n$

n = \lambda _1+\lambda _2+\cdots +\lambda _n, \quad \lambda _1 \geq \lambda _2 \geq \cdots \geq \lambda _n \geq 0 .

For example for $n=4$ we have the 5 partitions 4000 3100 2200 2110 1111 Let $k, n$ be positive integers. The number of partitions $p(1,n)$ can be found from the recursion

p(k,n) = \begin{cases} 0 & \text{if } k > n \\ 1 & \text{if } k = n \\ p(k+1,n)+p(k,n-k) & \text{otherwise} \end{cases}

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(i)

Give a C++ implementation of this recursion.

(ii)

Give a recursive implementation to find the partitions.

Problem 4.105

A number of Hamilton systems can be written in the form

\frac{dL}{dt} = [A,L](t)

where $L$ and $A$ are time-dependent $n \times n$ matrices. This is called the Lax representation of the Hamilton system. We find that

\frac{dL^k}{dt} = [A,L^k](t)

and the that $\tr [L^k]$ ( $k=1,2,\ldots$ ) are first integrals. Consider the Hamilton function (Toda lattice)

H(\overrightarrow {p},\overrightarrow {q}) = \frac{1}{2}(p_1^2+p_2^2+p_3^2) + \exp (q_1-q_2) + \exp (q_2-q_3) + \exp (q_3-q_1) .

Introducing the quantities

a_j := \frac{1}{2}\exp \left( \frac{1}{2}(q_j-q_{j+1}) \right), \quad b_j := \frac{1}{2}p_j

and cyclic boundary conditions (i.e., $q_4 \equiv q_1$ ) we find that the Hamilton equations of motion take the form (with $b_3 = 0$ )

\frac{da_j}{dt} = a_j(b_j-b_{j+1}), \quad \frac{db_1}{dt} = -2a_1^2, \quad \frac{db_2}{dt} = 2(a_1^2-a_2^2), \quad \frac{db_3}{dt} = 2a_2^2

where $j=1,2$ . Introducing the matrices (Lax pair)

L := \begin{pmatrix} b_1 & a_1 & 0 \\ a_1 & b_2 & a_2 \\ 0 & a_2 & b_3 \end{pmatrix}, \quad A := \begin{pmatrix} 0 & -a_1 & 0 \\ a_1 & 0 & -a_2 \\ 0 & a_2 & 0 \end{pmatrix}

the equations of motion can be written as Lax representation. From $L$ we find the first integral as $\tr [L^n]$ , where $n = 1,2,\ldots$ , where $\tr$ denotes the trace. We obtain

\tr L = b_1+b_2+b_3, \quad \tr [L^2] = b_1^2+b_2^2+b_3^2+2a_1^2+2a_2^2 .

Write a SymbolicC++ program lax.cpp that finds $[L,A]$ and shows that $\tr (L)$ , $\tr (L^2)$ and the determinant of $L$ are first integrals.

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Problem 4.106

The $n$ -qubit Pauli group is defined by

\mathcal{P}_n := \left\{ I_2, \sigma _x, \sigma _y, \sigma _z \right\} ^{\otimes n} \otimes \left\{ \pm 1, \pm i \right\} \tag {1}

where $\sigma_x, \sigma_y, \sigma_z$ are the $2\times 2$ Pauli matrices and $I_2$ is the $2\times 2$ identity matrix. The dimension of the Hilbert space under consideration is $\dim \mathcal{H} = 2^n$ . Thus each element of the Pauli group $\mathcal{P}_n$ is (up to an overall phase $\pm 1,\pm i$ ) a Kronecker product of Pauli matrices and $2\times 2$ identity matrices acting on $n$ qubits. The order of the Pauli group is $2^{2n+2}$ . Thus for $n=1$ we have the order 16. For $n=2$ we have order 64. Write a SymbolicC++ program that implements the Pauli group for $n=2$ .

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Problem 4.107

Let $b_1^\dagger , b_2^\dagger$ be Bose creation operator and let $I$ be the identity operator. The semi-simple Lie algebra $\mathfrak {su}(1,1)$ is generated by

K_+ := b_1^\dagger b_2^\dagger , \quad K_- := b_1b_2, \quad K_0 := \frac{1}{2}(b_1^\dagger b_1+b_2^\dagger b_2+I)

with the commutation relations

[K_0,K_+] = K_+, \quad [K_0,K_-] = -K_-, \quad [K_-,K_+] = 2K_0 .

We use the ordering $K_+, K_-, K_0$ for the basis. Write a computer algebra program that finds the ajoint representation for the Lie algebra.

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Problem 4.108

The exceptional Lie algebra $\mathfrak {g}_2$ has rank 2 and dimension 14. A basis is given by

H_1, H_2, X_1, X_2, X_3, X_4, X_5, X_6, Y_1, Y_2, Y_3, Y_4, Y_5, Y_6

with the commutation relations $[H_1,H_2] = 0$ and

H_1 = [X_1,Y_1], \quad H_2 = [X_2,Y_2], \quad [H_1,X_1] = 2X_1, \quad [H_2,X_2] = 2X_2 .

Thus

[H_1,Y_1] = -2Y_1, \quad [H_2,Y_2] = -2Y_2 .

Thus $H_1, X_1, Y_1$ and $H_2, X_2, Y_2$ each span the sub Lie algebra $\mathfrak {sl}(2,\mathbb {C})$ of $\mathfrak {g}_2$ . The commutator table for $\mathfrak {g}_2$ (Table 4.1) is, where entry $(i,j)$ gives $[\text{row}_i, \text{column}_j]$ for $i < j$ and is left blank (not part of the table) otherwise:

\begin{array}{c|ccccccccccccc}& H_2 & X_1 & Y_1 & X_2 & Y_2 & X_3 & Y_3 & X_4 & Y_4 & X_5 & Y_5 & X_6 & Y_6 \\ \hline H_1 & 0 & 2X_1 & -2Y_1 & -3X_2 & 3Y_2 & -X_3 & Y_3 & X_4 & -Y_4 & 3X_5 & -3Y_5 & 0 & 0 \\ H_2 & & -X_1 & Y_1 & 2X_2 & -2Y_2 & X_3 & -Y_3 & 0 & 0 & -X_5 & Y_5 & X_6 & -Y_6 \\ X_1 & & & H_1 & X_3 & 0 & 2X_4 & -3Y_2 & -3X_5 & -2Y_3 & 0 & Y_4 & 0 & 0 \\ Y_1 & & & & 0 & -Y_3 & 3X_2 & -2Y_4 & 2X_3 & 3Y_5 & -X_4 & 0 & 0 & 0 \\ X_2 & & & & & H_2 & 0 & Y_1 & 0 & 0 & -X_6 & 0 & 0 & Y_5 \\ Y_2 & & & & & & -X_1 & 0 & 0 & 0 & 0 & Y_6 & -X_5 & 0 \\ X_3 & & & & & & & H_1+3H_2 & -3X_6 & 2Y_1 & 0 & 0 & 0 & Y_4 \\ Y_3 & & & & & & & & -2X_1 & 3Y_6 & 0 & 0 & -X_4 & 0 \\ X_4 & & & & & & & & & 2H_1+3H_2 & 0 & -Y_1 & 0 & Y_3 \\ Y_4 & & & & & & & & & & X_1 & 0 & X_3 & 0 \\ X_5 & & & & & & & & & & & H_1+H_2 & 0 & -Y_2 \\ Y_5 & & & & & & & & & & & & X_2 & 0 \\ X_6 & & & & & & & & & & & & & H_1+2H_2 \end{array}

Note that $\mathfrak {sl}(3,\mathbb {C})$ is a sub Lie algebra of $\mathfrak {g}_2$ . Write a SymbolicC++ program that finds the adjoint representation.

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Problem 4.109

Let $b_1, b_2, b_3$ be Bose annihilation operators. Show that

H_1 = b_1^\dagger b_1 - b_2^\dagger b_2, \quad H_2 = b_2^\dagger b_2 - b_3^\dagger b_3

E_{12} = b_1^\dagger b_2, \quad E_{23} = b_2^\dagger b_3, \quad E_{13} = b_1^\dagger b_3

E_{21} = b_2^\dagger b_1, \quad E_{32} = b_3^\dagger b_2, \quad E_{31} = b_3^\dagger b_1

are a representation of the Lie algebra $\mathfrak {su}(3)$ . Write a SymbolicC++ program that implements this representation.

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Problem 4.110

Consider the fundamental representation of the superalgebra $\mathfrak {su}(2|1)$ . Let $\sigma_1, \sigma_2, \sigma_3$ be the Pauli spin matrices. Let $\oplus$ denote the direct sum. Its generators are given by the $3 \times 3$ matrices

L_1 = \frac{1}{2}\sigma _1\oplus (0), \quad L_2 = \frac{1}{2}\sigma _2\oplus (0), \quad L_3 = \frac{1}{2}\sigma _3\oplus (0), \quad L_4 = \frac{1}{2}\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix}

V_1 = \frac{1}{2}\begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}, \quad V_2 = \frac{1}{2}\begin{pmatrix} 0 & 0 & i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{pmatrix}

W_1 = \frac{1}{2}\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}, \quad W_2 = \frac{1}{2}\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix} .

Let

L_\pm = L_1\pm iL_2, \quad V_\pm = V_1\pm iV_2, \quad W_\pm = W_1\pm iW_2 .

Here $L_1, L_2, L_3, L_4$ are the generators forming the Lie subalgebra $\mathfrak {su}(2)\otimes \mathfrak {u}(1)$ of $\mathfrak {su}(2|1)$ and $V_1, V_2, W_1, W_2$ are the supergenerators. Let $c^\dagger , c$ be Fermi creation and annihilation operators. Let $b_1^\dagger , b_2^\dagger$ be Bose creation operators. A Boson-Fermion realization is given by

L_+ = b_1^\dagger b_2 \otimes I_F, \quad L_- = b_2^\dagger b_1 \otimes I_F

L_3 = \frac{1}{2}(b_1^\dagger b_1 \otimes I_F - b_2^\dagger b_2 \otimes I_F), \quad L_4 = \frac{1}{2}(b_1^\dagger b_1 \otimes I_F + b_2^\dagger b_2 \otimes I_F) + I_B\otimes c^\dagger c

V_+ = b_1^\dagger \otimes c, \quad V_- = b_1 \otimes c^\dagger , \quad W_+ = b_2^\dagger \otimes c, \quad W_- = b_2 \otimes c^\dagger .

Show that the representations are isomorphic. Give a SymbolicC++ implementation for the Bose-Fermi realization.

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