PhyslibAlpha.Mathematics.LadderSystem.Irreducibility

Given a configuration of occupancy numbers $\alpha: \{0, \dots, d-1\} \to \mathbb{N}$ (representing the number of quanta in each of $d$ modes) and indices $i, j \in \{0, \dots, d-1\}$, the function `moveOneTo` returns a new configuration $\alpha'$ representing the state after moving one quantum from mode $j$ to mode $i$. The resulting configuration is defined by: $$\alpha'(c) = \begin{cases} \alpha(i) + 1 & \text{if } c = i \\ \alpha(j) - 1 & \text{if } c = j \text{ and } c \neq i \\ \alpha(c) & \text{otherwise} \end{cases}$$ In the intended case where $i \neq j$, this corresponds to incrementing the occupancy of mode $i$ and decrementing the occupancy of mode $j$.

Let $d \in \mathbb{N}$ be the number of modes and $\alpha : \{0, \dots, d-1\} \to \mathbb{N}$ be a configuration representing the number of quanta in each mode. Let $\text{countWord}(d, \alpha)$ be the list of indices where each mode $c$ appears $\alpha(c)$ times. For any two distinct modes $i, j \in \{0, \dots, d-1\}$ such that $i \neq j$, the list formed by prepending $i$ to $\text{countWord}(d, \alpha)$ after removing one instance of $j$ is a permutation of the list $\text{countWord}(d, \alpha')$, where $\alpha'$ is the configuration obtained by moving one quantum from mode $j$ to mode $i$.

Let $V$ be a vector space over a field $K$ with a ladder system $L$ and a vacuum vector $\Omega \in V$. For any occupation number configuration $\alpha : \{0, \dots, d-1\} \to \mathbb{N}$, let the corresponding basis vector be denoted by $v_\alpha = L.\text{word}(\text{countWord}(d, \alpha), \Omega)$. For any two distinct modes $i, j \in \{0, \dots, d-1\}$, the action of the operator $E_{ij} = a^\dagger_i a_j$ on $v_\alpha$ is given by: $$E_{ij} v_\alpha = \alpha(j) \cdot v_{\alpha'}$$ where $\alpha'$ is the configuration obtained by moving one quantum from mode $j$ to mode $i$ (i.e., $\alpha'(i) = \alpha(i) + 1$, $\alpha'(j) = \alpha(j) - 1$, and $\alpha'(c) = \alpha(c)$ for $c \neq i, j$).

Let $V$ be a module over a field $K$, and let $W \subseteq V$ be a submodule. If $W$ is invariant under an endomorphism $M \in \text{End}_K(V)$ (meaning $M(w) \in W$ for all $w \in W$), then for any polynomial $p \in K[x]$, the submodule $W$ is also invariant under the endomorphism $p(M)$ obtained by evaluating $p$ at $M$. That is, for every $w \in W$, $p(M)w \in W$.

Let $V$ be a vector space over a field $K$ equipped with a ladder system $L$ of $d$ modes and a vacuum vector $\Omega$. Let $\text{vacuumSpan}(L, \Omega, n)$ be the $n$-th excitation-number sector, defined as the $K$-linear span of all vectors formed by applying $n$ creation operators to $\Omega$. Suppose $W \subseteq \text{vacuumSpan}(L, \Omega, n)$ is a nonzero submodule that is invariant under the action of the $\mathfrak{gl}(d)$ generators $E_{ij} = a^\dagger_i a_j$ for all $i, j \in \{0, \dots, d-1\}$. Then $W$ contains at least one occupation-number basis vector of the form $v_\alpha = a^\dagger_{i_1} a^\dagger_{i_2} \dots a^\dagger_{i_n} \Omega$ corresponding to a configuration $\alpha$ where $\sum_{c=0}^{d-1} \alpha(c) = n$.

Let $d$ be the number of modes and $\alpha: \{0, \dots, d-1\} \to \mathbb{N}$ be a count function representing the occupancy of each mode. The total occupation number, given by the sum $\sum_{c=0}^{d-1} \alpha(c)$, is equal to the length of the canonical word $\text{countWord}(d, \alpha)$ representing that configuration.

Let $\alpha: \{0, \dots, d-1\} \to \mathbb{N}$ be a configuration representing the occupancy numbers of $d$ modes. For any two distinct modes $i, j \in \{0, \dots, d-1\}$ such that the source mode $j$ is not empty ($\alpha(j) \neq 0$), the total occupation number remains invariant when one quantum is moved from mode $j$ to mode $i$: $$\sum_{c=0}^{d-1} (\text{moveOneTo}(\alpha, i, j))(c) = \sum_{c=0}^{d-1} \alpha(c).$$

Let $\alpha: \{0, \dots, d-1\} \to \mathbb{N}$ be a configuration representing the number of quanta in each of $d$ modes. For any two modes $i$ and $j$, let $\alpha' = \text{moveOneTo}(\alpha, i, j)$ be the configuration obtained by moving a quantum from mode $j$ to mode $i$. The occupancy of the target mode $i$ in the new configuration is given by $\alpha'(i) = \alpha(i) + 1$.

Let $\alpha: \{0, \dots, d-1\} \to \mathbb{N}$ be a configuration of occupancy numbers representing the number of quanta in each of $d$ modes. For any two distinct modes $i, j \in \{0, \dots, d-1\}$ where the occupancy of mode $j$ is non-zero ($\alpha(j) \neq 0$), moving a quantum from mode $j$ to mode $i$ and then immediately moving a quantum from mode $i$ back to mode $j$ returns the system to its original configuration $\alpha$. That is, $$\text{moveOneTo}(\text{moveOneTo}(\alpha, i, j), j, i) = \alpha$$

Let $V$ be a $K$-vector space with a ladder system of $d$ modes and a vacuum vector $\Omega$. Let $E_{ij} = a^\dagger_i a_j$ be the bilinear operators for $i, j \in \{0, \dots, d-1\}$. Suppose $W \subseteq V$ is a submodule invariant under all $E_{ij}$. Let $\beta: \{0, \dots, d-1\} \to \mathbb{N}$ be an occupancy configuration, and let $|\beta\rangle$ denote the state vector produced by applying the corresponding creation operators to $\Omega$. If $|\beta\rangle \in W$ and there exists a mode $k$ such that the occupancy $\beta(k) > 0$, then for any mode $l \neq k$, the state $|\beta'\rangle$ obtained by moving one quantum from mode $k$ to mode $l$ is also contained in $W$.

In a ladder system $L$ on a $K$-vector space $V$ with $d > 0$ modes and vacuum state $\Omega$, let $W \subseteq V$ be a submodule invariant under the $\mathfrak{gl}(d)$ generators $E_{ij} = a^\dagger_i a_j$ for all $i, j \in \{0, \dots, d-1\}$. If $W$ contains an occupation-number state $|\alpha\rangle$ (the state formed by applying creation operators to $\Omega$ according to the occupancy configuration $\alpha: \{0, \dots, d-1\} \to \mathbb{N}$), then $W$ must also contain the "hub" state $|(N, 0, \dots, 0)\rangle$, where $N = \sum_{c=0}^{d-1} \alpha(c)$ is the total number of excitations and all excitations are concentrated in mode $0$.

Let $V$ be a vector space over a field $K$ with a ladder system $L$ of $d > 0$ modes and a vacuum vector $\Omega \in V$. Let $W \subseteq V$ be a submodule invariant under the action of the bilinear operators $E_{ij} = a^\dagger_i a_j$ for all $i, j \in \{0, \dots, d-1\}$. If $W$ contains the "hub" state with total excitation number $n$, where all excitations are in the $0$-th mode: $$(a^\dagger_0)^n \Omega \in W,$$ then $W$ contains every state with the same total excitation number $n$. That is, for any occupation number distribution $\alpha: \{0, \dots, d-1\} \to \mathbb{N}$ such that $\sum_{c=0}^{d-1} \alpha(c) = n$, the corresponding state belongs to $W$: $$\left( \prod_{c=0}^{d-1} (a^\dagger_c)^{\alpha(c)} \right) \Omega \in W.$$

Let $L$ be a ladder system on a $K$-vector space $V$ with $d > 0$ modes and vacuum vector $\Omega$. Let $W$ be a submodule of $V$ that is invariant under the action of the bilinear operators $E_{ij} = a^\dagger_i a_j$ for all $i, j \in \{0, \dots, d-1\}$. Suppose $\alpha$ and $\alpha'$ are two occupation count functions from $\{0, \dots, d-1\}$ to $\mathbb{N}$ such that they have the same total excitation number: $$\sum_{c=0}^{d-1} \alpha(c) = \sum_{c=0}^{d-1} \alpha'(c)$$ If the state $w_\alpha = (a_0^\dagger)^{\alpha(0)} (a_1^\dagger)^{\alpha(1)} \dots (a_{d-1}^\dagger)^{\alpha(d-1)} \Omega$ is contained in $W$, then the state $w_{\alpha'} = (a_0^\dagger)^{\alpha'(0)} (a_1^\dagger)^{\alpha'(1)} \dots (a_{d-1}^\dagger)^{\alpha'(d-1)} \Omega$ is also contained in $W$.

Let $V$ be a vector space over a field $K$ with a ladder system $L$ of $d > 0$ modes and a vacuum vector $\Omega \in V$. Let $\text{vacuumSpan}(L, \Omega, n)$ denote the $n$-th excitation-number sector, defined as the $K$-linear submodule of $V$ spanned by all vectors of the form $a^\dagger_{i_1} a^\dagger_{i_2} \dots a^\dagger_{i_n} \Omega$. Suppose $W$ is a submodule of $\text{vacuumSpan}(L, \Omega, n)$ that is invariant under the action of the bilinear operators $E_{ij} = a^\dagger_i a_j$ for all $i, j \in \{0, \dots, d-1\}$. If $W$ is not the zero submodule ($W \neq \{0\}$), then $W$ must be the entire excitation-number sector: $$W = \text{vacuumSpan}(L, \Omega, n)$$ This implies that the $n$-th excitation-number sector is irreducible under the action of the $\mathfrak{gl}(d)$ generators $E_{ij}$.