Dependencies

Let \(T\) be a matrix product operator tensor and \(A_x\) a matrix product state tensor, and let \(T\cdot A_x\) be the tensor of the operator applied to the state, \((T\cdot A_x)^i=\sum _jT^{ij}\otimes A_x^j\). If the periodic operators of \(T\) map the periodic vectors of \(A_x\) to those of a normal tensor \(A_y\) of positive bond dimension at every positive length, then \(T\cdot A_x\) reduces to \(A_y\) with a nilpotent remainder. If, at every positive length, they map them to a finite weighted sum \(\sum _s\mu _s^N|V^{(N)}(A_s)\rangle \) over normal tensors of positive bond dimension with nonzero weights \(\mu _s\), then \(T\cdot A_x\) admits a simultaneous reduction to the weighted blocks \(\mu _sA_s\).

Let \(\{ M_c\} _c\) be matrix product operator tensors of positive bond dimensions \(\chi _c\), each normal when its pair of physical indices is treated as one index, with \(O_N(M_a)O_N(M_b)=\sum _cN_{ab}^cO_N(M_c)\) for every \(N\geq 1\). Then the stacked product of \(M_a\) and \(M_b\) has fusion tensors \(V_{ab}^{c,\mu }\), \(W_{ab}^{c,\mu }\), \(\mu {\lt}N_{ab}^c\), as in Theorem 29.6.1.1, with nilpotency order at most \(\chi _a\chi _b\). Likewise, if \(O_N(T)|V^{(N)}(A_x)\rangle =\sum _yM_y|V^{(N)}(A_y)\rangle \) for every \(N\geq 1\) with normal \(A_y\) of positive bond dimension, then \(T\cdot A_x\) has action tensors onto \(M_y\) copies of each \(A_y\).

Let \((V,W)\) and \((V',W')\) be two reductions of the stacked product \(B\) of \(M_a\) and \(M_b\) onto the same normal tensor \(M_c\), each with a remainder all of whose products of at least \(K\) factors vanish. Then there is \(z\neq 0\) with \(VB^{w}=z\, V'B^{w}\) and \(B^{w}W=z^{-1}B^{w}W'\) for every word \(w\) of length greater than \(2K\). When \(O_N(M_a)O_N(M_b)=O_N(M_c)\) for every \(N\geq 1\), Corollary 29.6.1.2 provides such reductions with \(K=\chi _a\chi _b\).

Let \(T_a\) be a matrix product operator tensor, \(A_x\) a tensor, and \(A_y\) normal tensors of positive bond dimension that are pairwise not gauge related. Two families of action tensors that split \(T_a\cdot A_x\) over the blocks \(A_y\) with multiplicities \(M_{a,x}^y\), as in (13), differ by a unique \(X_{ax}^y\in \mathrm{GL}_{M_{a,x}^y}(\mathbb {C})\) for each \(y\) in the sense of (14). This is the gauge freedom of action tensors in [ 199 , Section 3.1 ] .

Let \(T_a\), \(T_b\) and \(T_c\) be matrix product operator tensors, with the \(T_c\) normal as tensors on the pair alphabet, of positive bond dimension, and pairwise not gauge related. Two families of fusion tensors that split the stacked product \(T_aT_b\) over the blocks \(T_c\) with multiplicities \(N_{ab}^c\), as in (13), differ by a unique \(Y_{ab}^c\in \mathrm{GL}_{N_{ab}^c}(\mathbb {C})\) for each \(c\) in the sense of (14).

Let \(B\) be a tensor, \(C\) a normal tensor, and \((V,W)\), \((\widetilde V,\widetilde W)\) two reductions of \(B\) onto \(C\) whose residual tensors \(B-WCV\) and \(B-\widetilde WC\widetilde V\) are nilpotent with a common bound \(N_0\). Then there is a unique \(\lambda \in \mathbb {C}^\times \) with

for every word with \(|\mathbf w|{\gt}2N_0\). This applies to the fusion tensors of a group-like algebra, where \(T_gT_h\) reduces onto the single block \(T_{gh}\), to the composite reductions of Theorem 32.3.1, and to the action tensors of an invertible label on a permuted block. It does not apply to a single slot of a reduction onto multiple blocks, whose residual is not nilpotent.

Let \(\Lambda \) be a finite set. A complete zipper fusion family consists of positive bond dimensions \(D_a\), MPO blocks \(B_a^{ij}\in M_{D_a}(\mathbb {C})\), fusion multiplicities \(N_{ab}^{c}\), and matrices

The matrices \(X^{c+}_{ab,\mu }\) are independently chosen left inverses, not adjoints. For all labels and multiplicity indices,

The opposite product \(P_{ab}=\sum _{c,\mu }X^c_{ab,\mu }X^{c+}_{ab,\mu }\) is the projector onto the product-MPO support; it need not be the identity on the ambient bond space or an orthogonal projector. Write \(B_{ab}^{ik}=\sum _j B_a^{ij}\otimes B_b^{jk}\). Every product-MPO matrix is reconstructed on this support:

The two zipper identities are

Each block is injective, and the matrix of coefficients of all labelled blocks has a simultaneous left inverse. Contraction of its row indexed by \((c,x,y)\) with \(B_d^{ij}\) gives \(E_{xy}\) when \(d=c\) and zero otherwise. These are the fusion-tensor, reconstruction, zipper, and simultaneous inverse relations of [ 8 , lines 161, 181–200, and 269–277 ] .

A finite family \((T_a)_{a\in I}\) of matrix product operator tensors with structure constants \(N_{ab}^c\in \mathbb {N}\) is a fusion algebra if, for all \(a,b\in I\) and \(N\geq 1\),

This is the fusion rule of [ 199 , Section 2 ] and [ 8 ] .

A finite family \((A_x)_{x\in K}\) of matrix product state tensors is symmetric under \((T_a)_{a\in I}\) with coefficients \(M_{a,x}^y\in \mathbb {C}\) if, for all \(a\), \(x\) and \(N\geq 1\),

A single tensor \(A\) is symmetric with eigenvalues \(c_a\) if \(O_a^{(N)}|V^{(N)}(A)\rangle =c_a|V^{(N)}(A)\rangle \) for all \(a\) and \(N\geq 1\). The coefficients do not depend on \(N\).

Let \((A_x)_{x\in K}\) be a symmetric family of normal tensors of positive bond dimension whose coefficients \(M_{a,x}^y\) are nonnegative integers. A family of action tensors for \((a,x)\) consists of maps \(V_{ax}^{y,i}\colon \mathbb {C}^{\chi _a}\otimes \mathbb {C}^{D_x}\to \mathbb {C}^{D_y}\) and \(W_{ax}^{y,i}\colon \mathbb {C}^{D_y}\to \mathbb {C}^{\chi _a}\otimes \mathbb {C}^{D_x}\), for \(y\in K\) and \(1\leq i\leq M_{a,x}^y\), that are mutually biorthogonal reductions of \(T_a\cdot A_x\) onto the blocks \(A_y\) with a nilpotent remainder.

The associator of the chosen fusion tensors is the function \(\omega \colon G^3\to \mathbb {C}^\times \) defined by

for all sufficiently long words \(\mathbf w\); equivalently, \((T_gT_hT_k)^{\mathbf w}W^{g(hk)}=\omega (g,h,k)\, (T_gT_hT_k)^{\mathbf w}W^{(gh)k}\) for the composite embeddings, which is the orientation of the defining display of [ 197 ] . It exists and is nonzero by Corollary 32.2.1 applied to the two composite reductions of Theorem 32.3.1, and it is unique because \(T_{ghk}^{\mathbf w}\neq 0\) for some long word. This is the three-cocycle of [ 199 , Section 4.4 ] , and it is the anomaly three-cochain of Definition 19.5.1.3, characterized by Theorem 19.5.1.5.

Let \((T_a)_{a\in I}\) be a fusion algebra of normal tensors of positive bond dimension. A family of fusion tensors for the pair \((a,b)\) consists of maps

for \(c\in I\) and \(1\leq \mu \leq N_{ab}^c\), such that the pairs are mutually biorthogonal, \(V_{ab}^{c,\mu }W_{ab}^{d,\nu }=\delta _{cd}\delta _{\mu \nu } \mathbb {1}_{\chi _c}\), each pair is a reduction of \(T_aT_b\) onto \(T_c\),

for every word \(\mathbf w\) in the pair alphabet, and the remainder \((T_aT_b)^{ij}-\sum _{c,\mu }W_{ab}^{c,\mu }T_c^{ij}V_{ab}^{c,\mu }\) generates a nilpotent algebra. This is the decomposition of [ 199 , Section 2 ] and [ 8 ] , with the nilpotent remainder of [ 199 , Section 4.4 ] in place of an exact identity.

In the setting of Section 32.3, let \((A_x)_{x\in K}\) be a symmetric family as in Theorem 30.15.11, so that \(O_g^{(N)}|V^{(N)}(A_x)\rangle =|V^{(N)}(A_{g\cdot x})\rangle \), with action tensors \((V_{g,x},W_{g,x})\) reducing \(T_g\cdot A_x\) onto \(A_{g\cdot x}\). The tensor \(T_gT_h\cdot A_x\) has two composite reductions onto \(A_{gh\cdot x}\), and the \(L\)-symbol \(L^x_{g,h}\in \mathbb {C}^\times \) is defined by

for all sufficiently long words \(w\). This is the \(L\)-symbol of [ 199 , Section 4.4 ] .

Let \(A\) and \(B\) have the same physical alphabet and respective bond dimensions \(D_A\) and \(D_B\). A reduction from \(B\) to \(A\) is a pair \(V\in \mathbb {C}^{D_A\times D_B}\) and \(W\in \mathbb {C}^{D_B\times D_A}\) such that

for every word \(\mathbf a\), including the empty word. This is the reduction of [ 221 , Proposition 20 and the following definition ] .

A natural number \(L\) is a residual-word nilpotency bound if every word \(N^{i_1}\cdots N^{i_L}\) of exactly \(L\) residual letters is zero. The residual nilpotency length \(\ell (B,A;V,W)\) is the infimum of the set of exact-length bounds. Definition 8 of [ 221 ] instead asks that every residual word of every length \(n\geq L\) vanish; the passage from an exact-length bound to this consequence is stated separately in Theorem 29.11.21.

Let \(\{ O_g\} _{g\in G}\) be matrix product operator tensors of positive bond dimension whose doubled-index tensors are normal and whose periodic operators satisfy \(O_g O_h=O_{gh}\) on every nonempty chain. A choice of fusion tensors is, for every pair \(g,h\), a reduction \((F^{\lt}_{g,h},F^{\gt}_{g,h})\) of \(O_gO_h\) onto \(O_{gh}\). The triple product \(T=(O_gO_h)O_k\) reduces onto \(O_{ghk}\) along the two fusion trees with left boundaries

where the target bond space of \(F^{\lt}_{g,hk}\), that of \(O_{g(hk)}\), is identified with that of \(O_{(gh)k}\) through the equality \(g(hk)=(gh)k\) in \(G\); the same identification enters the right boundaries \(W^L=(F^{\gt}_{g,h}\otimes \mathbf1)F^{\gt}_{gh,k}\) and \(W^R=\mathsf a(\mathbf1\otimes F^{\gt}_{h,k})F^{\gt}_{g,hk}\). The anomaly three-cochain value \(\omega (g,h,k)\in \mathbb {C}^\times \) is the scalar with \(V^LT^{\mathbf w}=\omega (g,h,k)\, V^RT^{\mathbf w}\) for all long words \(\mathbf w\). Equivalently \(T^{\mathbf w}W^R=\omega (g,h,k)\, T^{\mathbf w}W^L\) for the right boundaries, which is the defining display of [ 197 ] before its three-cocycle equation. No normalization of \(\omega \) is imposed.

Two scalar \(3\)-cochains \(\omega \) and \(\eta \) are cohomologous when \(\omega =\beta \mathbin {\triangleright }\eta \) for some scalar \(2\)-cochain \(\beta \).

For a scalar \(2\)-cochain \(\beta \), define

Formula (396) is the change of \(\omega \) induced by (262), as in  [ 197 ] .

An action-tensor scalar gauge is a function \(\gamma \colon G\times \mathsf X\to \mathbb {C}^\times \), written \(\gamma ^x_g\). Together with a fusion-tensor scalar gauge \(\beta \colon G^2\to \mathbb {C}^\times \), it changes an \(L\)-symbol by

This is the joint fusion-tensor and action-tensor gauge freedom in [ 197 ] , written with the block label as a superscript throughout, as that source does for its block-dependent cochains.

Let \(G\) act on a set \(\mathsf X\). An \(L\)-symbol is a function \(L\colon \mathsf X\times G^2\to \mathbb {C}^\times \), written \(L^x_{g,h}\). It is compatible with a scalar \(3\)-cochain \(\omega \) when, for every \(x\in \mathsf X\) and \(g,h,k\in G\),

The \(L\)-symbols were introduced by [ 199 ] ; this is the scalar compatibility relation of [ 197 ] . No transitivity or finiteness assumption is imposed on the action.

A scalar \(3\)-cochain is a \(3\)-cocycle when, for all \(g,h,k,l\in G\),

This is the scalar \(3\)-cocycle relation of  [ 197 ] .

A scalar \(3\)-cochain has trivial gauge class when it is cohomologous to the constant cochain \(1\).

A tensor \(A\) is normal [ 17 ] if there exists a positive integer \(L_0\) such that \(A\) is \(L_0\)-block injective.

Let \(\{ M_\gamma \} _\gamma \) be a finite family of matrix product operator tensors of positive bond dimension, each normal when its pair of physical indices is treated as one index. Suppose that nonzero scalars \(\lambda _\gamma \) satisfy \(O_N(B)=\sum _\gamma \lambda _\gamma ^NO_N(M_\gamma )\) for every \(N\geq 1\). Then for every \(\gamma \) there are maps \(V_\gamma \) and \(W_\gamma \) with \(W_\gamma V_\gamma =\operatorname{id}\) and \(W_\gamma B^{i_1j_1}\cdots B^{i_Nj_N}V_\gamma =\lambda _\gamma ^NM_\gamma ^{i_1j_1}\cdots M_\gamma ^{i_Nj_N}\) for every \(N\geq 1\) and every sequence of index pairs. The pairs may be chosen mutually biorthogonal, and the remainder after all matched sectors are removed is nilpotent.

Let \(\{ C_c\} _c\) be a finite family of normal tensors of positive bond dimension and let \(N_c\geq 0\) be integers. If a tensor \(B\) of bond dimension \(D\) satisfies \(|V^{(N)}(B)\rangle =\sum _cN_c|V^{(N)}(C_c)\rangle \) for every \(N\geq 1\), then for every pair \((c,\mu )\) with \(\mu {\lt}N_c\) there are maps \(V_{c,\mu }\) and \(W_{c,\mu }\) such that \(V_{c,\mu }W_{d,\nu }=\delta _{cd}\delta _{\mu \nu }\operatorname{id}\), \(V_{c,\mu }B^{i_1}\cdots B^{i_N}W_{c,\mu }=C_c^{i_1}\cdots C_c^{i_N}\) for every \(N\geq 0\), and every product of at least \(D\) matrices of the remainder \(B^i-\sum _{c,\mu }W_{c,\mu }C_c^iV_{c,\mu }\) vanishes.

Let the blocks \(C_c\) be normal of positive bond dimension and pairwise not gauge related in the sense of Definition 29.8.1.1, and let \((V,W)\) and \((\widetilde V,\widetilde W)\) be two split compressions of \(B\) as in (13) with the same blocks and multiplicities. Then \(\widetilde V_{c,\mu }W_{e,\nu }=0\) for \(e\neq c\), and \(\widetilde V_{c,\mu }W_{c,\nu }=(Y_c)_{\mu \nu }\operatorname{id}\) for a matrix \(Y_c\in \mathrm{GL}_{N_c}(\mathbb {C})\) whose inverse is given by \(V_{c,\mu }\widetilde W_{c,\nu }=(Y_c^{-1})_{\mu \nu }\operatorname{id}\). The matrices \(Y_c\) are the unique ones with

for every physical index \(i\), so the two compressions differ by \(\bigoplus _c\mathrm{GL}_{N_c}(\mathbb {C})\) on the support of \(B\), with no dressing by words. This is the uniqueness of fusion tensors under the zipper condition in [ 8 ] .

A fusion character with values in \(\mathbb {N}\) is \(1\) on every invertible label, so an invertible label fixes the periodic vectors of a single symmetric normal tensor with \(c_e=1\). For a family as in Theorem 30.15.8 on which \(O_e^{(N)}\) fixes every periodic vector, each invertible label \(g\) permutes the blocks: there is a bijection \(x\mapsto g\cdot x\) of the block labels with \(O_g^{(N)}|V^{(N)}(A_x)\rangle =|V^{(N)}(A_{g\cdot x})\rangle \) for all \(N\geq 1\). This is the permutation action of [ 199 , Section 4 ] .

Let \((A_x)_{x\in K}\) be normal tensors of positive bond dimension whose periodic vectors are linearly independent at some length \(N_0\geq 1\), symmetric under a fusion algebra with coefficients \(M_{a,x}^y\). Then every \(M_{a,x}^y\) is a nonnegative integer and (100) holds. This is the integrality and associativity of the action multiplicities in [ 199 , Section 3 ] .

Under the hypotheses of Theorem 30.15.8, every pair \((a,x)\) admits a family of action tensors; for integer coefficients this is the action clause of Corollary 29.6.1.2. This is the existence of action tensors in [ 199 , Section 3 ] .

The associator satisfies the three-cocycle relation (394),

which is the pentagon equation for a group-like algebra [ 199 , Section 4 ] . This is Theorem 19.5.1.6.

Replacing the fusion tensors by \((\beta _{g,h}^{-1}V_{g,h}, \beta _{g,h}W_{g,h})\), so that \(V_{g,h}B^{\mathbf w}=\beta _{g,h}\, V'_{g,h}B^{\mathbf w}\) for the new compression \(V'_{g,h}\), with \(\beta \colon G^2\to \mathbb {C}^\times \) replaces \(\omega \) by \(\beta \mathbin {\triangleright }\omega \) of Definition 19.5.10.32. Consequently the class of \(\omega \) under the relation of Definition 19.5.10.37 does not depend on the choice of fusion tensors, and depends only on the family \((T_g)_{g\in G}\). This is the coboundary freedom of [ 199 , Section 4 ] , and it is Theorem 19.5.1.7.

Let \((V_1,W_1)\) be a reduction of a tensor \(C\) onto \(B\) whose residual letters \(N^i=C^i-W_1B^iV_1\) have vanishing products of length \(N_1\), and let \((V_2,W_2)\) be a pair of matrices, not necessarily a reduction of \(B\) onto \(A\), whose residual letters \(M^i=B^i-W_2A^iV_2\) have vanishing products of length \(N_2\). Then the residual letters \(R^i=C^i-W_1W_2A^iV_2V_1\) of the composite pair \((V_2V_1,W_1W_2)\) have vanishing products of length \(2N_1+N_2\). The residual bound is unchanged when a reduction is tensored with an identity strand, \((V\otimes \mathbb {1},W\otimes \mathbb {1})\) on \(T_aT_d\) or \((\mathbb {1}\otimes V,\mathbb {1}\otimes W)\) on \(T_dT_a\), and likewise for an operator tensor acting on a state tensor, and it is unchanged under the bond associator. In particular, if \((V,W)\) reduces \(T_aT_b\) onto \(T_c\) and \((V',W')\) reduces \(T_cT_d\) onto \(T_e\), both with residual bound \(K\), then \((V'(V\otimes \mathbb {1}),(W\otimes \mathbb {1})W')\) reduces \(T_aT_bT_d\) onto \(T_e\) with residual bound \(3K\). The reduction property, without the residual bound, is Lemma 19.5.1.1, and for the two fusion trees of a triple product it is Lemma 19.5.1.4.

Let \((B_a)_{a\in \Lambda }\) be a complete zipper fusion family (Definition 21.3.85) with fusion tensors \(X^c_{ab,\mu }\) and left inverses \(X^{c+}_{ab,\mu }\). For \(X=(X_c)_c\in \bigoplus _cM_{D_c}(\mathbb {C})\) put

Then the elements \((B_c^{ik})_c\) span \(\bigoplus _cM_{D_c}(\mathbb {C})\), and

The map \(\Delta \) is the only linear map with (14), so it depends only on the blocks. It is multiplicative, \(\Delta (XY)=\Delta (X)\Delta (Y)\), and \(\Delta (\mathbb {1})_{ab}=\sum _{c,\mu }X^c_{ab,\mu }X^{c+}_{ab,\mu }\) is the support projector of the fusion tensors. In the notation of [ 222 , Section 4 ] , \(X^c_{ab,\mu }\) and \(X^{c+}_{ab,\mu }\) are the source’s \(V_{ab}^{c\mu }\) and \(W_{ab}^{c\mu }\), (13) is the decomposition of the coproduct of the dual algebra, (14) is the coproduct of the matrix coefficients of the physical representation, and multiplicativity is the pre-bialgebra axiom.

The coproduct (13) is coassociative, \((\Delta \otimes \mathbb {1})\circ \Delta =(\mathbb {1}\otimes \Delta )\circ \Delta \), where on the triple bond space

This is the coassociativity axiom of a coalgebra [ 222 , Section 3 ] . The source assumes it and derives the associativity of fusion trees and the \(F\)-symbols; here it is proved from the fusion tensors.

Every pair \((a,b)\) of labels of a fusion algebra of normal tensors of positive bond dimension admits a family of fusion tensors (Corollary 29.6.1.2). This is the periodic-trace reduction under the fusion rule (96). It allows a nilpotent remainder and does not assert the exact local reconstruction of [ 199 , Appendix A ] . The exact fusion and action decompositions from length-independent arbitrary-boundary hypotheses are proved in Theorem 30.2.8.

The \(L\)-symbols of Definition 32.4.1 are compatible with the associator in the sense of (417),

and replacing the fusion and action tensors by \((\beta _{g,h}^{-1}V_{g,h},\beta _{g,h}W_{g,h})\) and \(((\gamma ^x_g)^{-1}V_{g,x},\gamma ^x_gW_{g,x})\) changes \((\omega ,L)\) to \((\beta \mathbin {\triangleright }\omega ,(\beta ,\gamma )\mathbin {\triangleright }L)\) of Definitions 19.5.10.32 and 19.5.10.50. This is the coupled pentagon equation for a group-like algebra [ 199 , Section 4 ] .

Let \((T_g)_{g\in G}\) be a group-like fusion algebra of normal tensors whose associator does not have trivial gauge class. Then there is no normal tensor \(A\) of positive bond dimension with \(O_g^{(N)}|V^{(N)}(A)\rangle =|V^{(N)}(A)\rangle \) for all \(g\in G\) and \(N\geq 1\). This is the statement of [ 199 , Section 3.2 ] that an injective state without multiplicity cannot be invariant under a nontrivial algebra, specialized to group-like algebras; for group-like algebras it goes back to [ 11 ] .

Let the blocks \(A_y\) be normal and pairwise not related by a gauge transformation. Two families of action tensors for \((a,x)\) with vanishing remainder, \((T_a\cdot A_x)^w=\sum _{y,i}W_{ax}^{y,i}A_y^w V_{ax}^{y,i}\) for every nonempty word, are related by a unique \(X_{ax}^y\in \mathrm{GL}_{M_{a,x}^y}(\mathbb {C})\) for each \(y\), in the sense of Theorem 32.2.2. This is the gauge freedom of the action tensors in [ 199 , Section 3.1 ] , and it is Corollary 29.8.1.5.

Let the blocks \(T_c\) of a fusion algebra be normal and pairwise not related by a gauge transformation, and let \((V_{ab}^{c,\mu },W_{ab}^{c,\mu })\) and \((\widetilde V_{ab}^{c,\mu },\widetilde W_{ab}^{c,\mu })\) be two families of fusion tensors for \((a,b)\) with vanishing remainder,

for every nonempty word \(\mathbf w\). Then for each \(c\) there is a unique \(Y_{ab}^c\in \mathrm{GL}_{N_{ab}^c}(\mathbb {C})\) with \(\widetilde V_{ab}^{c,\mu }=\sum _\nu (Y_{ab}^c)_{\mu \nu }V_{ab}^{c,\nu }\) and \(\widetilde W_{ab}^{c,\mu }=\sum _\nu W_{ab}^{c,\nu } \bigl((Y_{ab}^c)^{-1}\bigr)_{\nu \mu }\) on the support of the decomposition, so the ambiguity is \(\bigoplus _c\mathrm{GL}_{N_{ab}^c}(\mathbb {C})\) with no word dressing. This is the uniqueness of fusion tensors under the zipper condition in [ 8 ] , and the basis change of [ 222 , Section 4 ] ; it is Corollary 29.8.1.4.

Let \(A\) be normal, let \(B\) have the same physical alphabet, and let \((V,W)\) and \((\widetilde V,\widetilde W)\) be two reductions from \(B\) to \(A\). Suppose that \(N_0\in \mathbb N\) is a residual-word nilpotency bound for both reductions: every word of length \(N_0\) in either residual tensor vanishes. Then there exists a single scalar \(\lambda \in \mathbb {C}\) with \(\lambda \ne 0\) such that, for every word \(\mathbf a\) satisfying

one has

If \(A\) and \(B\) have the same positive-length closed-chain coefficients and the two least residual nilpotency lengths are at most \(N_0\), then \(N_0\) is such a common bound and the same conclusion follows. The direct-bound statement itself requires no equality of closed-chain coefficients.

Following [ 221 , Theorem 22 ] , these boundary-dressed identities hold for \(|\mathbf a|{\gt}2N_0\).

The scalar \(\omega (g,h,k)\) exists, is nonzero, and is the only scalar with the dressed identity of Definition 19.5.1.3.

For every \(g,h,k,l\in G\),

Let \(\omega \) and \(\omega '\) come from two choices of fusion tensors, and let \(\beta _{g,h}\in \mathbb {C}^\times \) be the scalar with \(F^{\lt}_{g,h}B^{\mathbf w}=\beta _{g,h}F'^{\lt}_{g,h}B^{\mathbf w}\) for all long words of \(B=O_gO_h\). Then

so the cohomology class of \(\omega \) does not depend on the fusion tensors, as in (262).

Let \(A\) be normal with positive bond dimension, and let \((V,W)\) and \((V',W')\) reduce \(B\) onto \(A\), where \(B\) and \(A\) generate the same matrix product vectors on nonempty chains. Then there is a unique \(z\in \mathbb {C}^\times \) such that \(VB^{\mathbf w}=z\, V'B^{\mathbf w}\) for all sufficiently long words \(\mathbf w\). If \(XA^{\mathbf w}=z\, YA^{\mathbf w}\) for all long words, then \(XVB^{\mathbf w}=z\, YVB^{\mathbf w}\) for all long words.

Let \((V,W)\) be a reduction from \(B\) to \(A\) and let \(\beta \in \mathbb {C}^\times \). Then \((\beta ^{-1}V,\beta W)\) is again a reduction from \(B\) to \(A\); its residual letters \(N^i=B^i-WA^iV\) are unchanged, hence so are its residual-word nilpotency bounds and its residual nilpotency length \(\ell (B,A;V,W)\); and a natural number is an exterior buffer length for \((\beta ^{-1}V,\beta W)\) if and only if it is one for \((V,W)\).

Consequently, let \(\beta \colon G^2\to \mathbb {C}^\times \) be a scalar \(2\)-cochain and let \(\{ \mathcal U_g\} _{g\in G}\) have a family of selected reductions. The rescaled matrices of (262), namely \(W_{g,h}\mapsto \beta _{g,h}W_{g,h}\) and \(V_{g,h}\mapsto \beta _{g,h}^{-1}V_{g,h}\), again form a family of selected reductions of \(\{ \mathcal U_g\} _{g\in G}\). It has exterior buffer length \(m\) exactly when the original family does, and rescaling commutes with blocking by any length \(L\).

This is the scalar gauge freedom of the fusion tensors in [ 197 , lines 1500–1504 ] under the identification (163); uniqueness up to boundary dressing is given by Theorem 29.11.28.

The pairs \((V^L,W^L)\) and \((V^R,W^R)\) of Definition 19.5.1.3 are reductions of the triple product \((O_gO_h)O_k\) onto \(O_{ghk}\).

Reductions compose: if \((V_1,W_1)\) reduces \(C\) onto \(B\) and \((V_2,W_2)\) reduces \(B\) onto \(A\), then \((V_2V_1,W_1W_2)\) reduces \(C\) onto \(A\). If \((V,W)\) reduces the doubled-index tensor of \(X\) onto that of \(A\), then \((V\otimes \mathbf1,W\otimes \mathbf1)\) reduces \(XP\) onto \(AP\) and \((\mathbf1\otimes V,\mathbf1\otimes W)\) reduces \(PX\) onto \(PA\). A reduction \((V,W)\) of \(M(NP)\) gives the reduction \((V\mathsf a^{-1},\mathsf aW)\) of \((MN)P\), and conversely.