Write \(R^{ik}=\sum _jP^{ij}\otimes Q^{jk}\) and \((T\cdot A)^i=\sum _jT^{ij}\otimes A^j\). For all boundary matrices and all lengths, including zero,

Let \(U,V\) be matrices on \(\mathbb {C}^{D_1}\otimes \mathbb {C}^{D_2}\). They are equal if and only if \(\operatorname{tr}((X\otimes Y)U)=\operatorname{tr}((X\otimes Y)V)\) for all matrices \(X,Y\) on the two factors.

Closedness is equivalent to the existence of a linear map \(b:M_{\chi ^2}(\mathbb {C})\to M_{\chi }(\mathbb {C})\) satisfying \(O^N_{TT,C}=O^N_{T,b(C)}\) for every \(C\) and \(N\geq 1\). Compatibility is equivalent to the existence of a linear map \(b:M_{\chi D}(\mathbb {C})\to M_{D}(\mathbb {C})\) satisfying \(\psi ^N_{T\cdot A,C}=\psi ^N_{A,b(C)}\) for every \(C\) and \(N\geq 1\). These are the boundary maps at the start of [ 199 , Appendix A ] .

Under (6) and (7), each nonempty word \(w\) satisfies

In particular, an exact fusion decomposition of \(PQ\) yields

for \(N\geq 1\), where the second identity uses an exact action decomposition of \(T\cdot A\) into the tensors \(B_c\).

If \(TT\) has an exact biorthogonal decomposition into copies of \(T\), then \(T\) is boundary closed. If \(T\cdot A\) has such a decomposition into copies of \(A\), then \(T\) and \(A\) are boundary compatible. In either case the output boundary is \(Z=\sum _cV_c(X\otimes Y)W_c\).

If in addition \(\sum _cW_cV_c=\mathbb {1}_D\), then \(B^w=\sum _cW_cA_c^wV_c\) also holds for the empty word.

Let \(A_c\) be individually injective tensors with \(D_c{\gt}0\), no two of which are equivalent by an invertible similarity and a nonzero scalar. Then some \(L{\gt}0\) satisfies \(\operatorname {span}\{ A^w:|w|=L\} =\mathcal D\).

Suppose \(b\) satisfies \(\operatorname{tr}(XB^w)=\sum _c\operatorname{tr}(b(X)_cA_c^w)\) for every \(X\) and every nonempty word \(w\). If \(\operatorname {span}\{ A^w:|w|=L\} =\mathcal D\) for some \(L{\gt}0\), its trace dual obeys

The map \(\rho \) need not preserve the ambient identity.

Every idempotent \(P\in M_D(K)\) over a field factors as \(P=WV\), \(VW=I_m\), with \(m=\operatorname {rank}P\). This includes \(m=0\) and imposes no self-adjointness condition.

A multiplicative \(K\)-linear map \(\rho :\prod _c M_{D_c}(K)\to M_D(K)\) admits finite multiplicities \(m_c\) and matrices \(W_{c\mu }:K^{D_c}\to K^D\), \(V_{c\mu }:K^D\to K^{D_c}\) such that

Zero-dimensional factors and zero multiplicities are allowed.

A length-independent block boundary map with a positive simultaneous spanning length gives an exact biorthogonal decomposition \(B^i=\sum _{c,\mu }W_{c\mu }A_c^iV_{c\mu }\). Consequently, arbitrary-boundary closedness gives such a decomposition of \(TT\), and arbitrary-boundary compatibility gives one of \(T\cdot A\), when the target tensor is the unweighted direct sum of these blocks.

If \(B^iW=WC^i\) and \(VW=I\), the boundary \(X\) of \(C\) is represented by \(WXV\) on \(B\) at every length. Applying this to the coordinate inclusions of the incoming blocks gives, for each pair \(a,b\), exact fusion tensors of \(T_aT_b\), and for each pair \(a,x\), exact action tensors of \(T_a\cdot A_x\), with their respective biorthogonality relations. Only the target block family requires a simultaneous spanning length.

For an unweighted block-diagonal MPO with individually injective, positive-dimensional, pairwise gauge-scalar inequivalent blocks, arbitrary-boundary closedness gives exact biorthogonal fusion tensors for every incoming pair \(a,b\), with finite multiplicities \(N_{ab}^c\):

Here \(W_{ab}^{c\mu }:\mathbb C^{\chi _c}\to \mathbb C^{\chi _a}\otimes \mathbb C^{\chi _b}\) and \(V_{ab}^{c\mu }\) has the reverse direction; when \(c\ne e\), the right side of (17) is the zero rectangular matrix. For an operator and a state in unweighted block-diagonal form, arbitrary-boundary compatibility gives exact biorthogonal action tensors for every incoming pair \(a,x\) if the state blocks satisfy those injectivity, dimension, and separation assumptions. With finite multiplicities \(M_{ax}^y\), they obey

The maps have the analogous directions, with incoming bond space \(\mathbb C^{\chi _a}\otimes \mathbb C^{D_x}\) and outgoing bond space \(\mathbb C^{D_y}\). These are the existence and orthogonality assertions of [ 199 , Appendix A ] .

Write \(B_{ab}^{ik}=\sum _j T_a^{ij}\otimes T_b^{jk}\) and \(C_{ab}^{ik}=\bigoplus _c(I_{N_{ab}^c}\otimes T_c^{ik})\). Exact biorthogonal decomposition implies

The product \(S_{ab}H_{ab}\) may be a proper, non-self-adjoint idempotent.

Replacing every letter by a common nonempty word preserves an exact biorthogonal decomposition and its rectangular maps. In particular, physical blocking preserves the multiplicities \(N_{ab}^c\) and both \(V_{ab}^{c\mu }\) and \(W_{ab}^{c\mu }\). For positive-dimensional normal blocks with no scalar-gauge duplicates, some common positive blocking gives a complete zipper fusion family with these same maps.

Under the block and arbitrary-boundary closedness assumptions of Theorem 30.2.8, there are exact unblocked fusion maps, finite multiplicities, and a complete zipper family after one positive physical blocking. Its blocks are precisely the blocked source tensors, and its synthesis, analysis, and multiplicities are the original unblocked ones.

For separated, positive-dimensional injective targets, two exact biorthogonal decompositions of the same tensor over these targets have identical multiplicities.

Two exact biorthogonal decompositions over the same multiplicities and separated positive-dimensional injective target blocks differ by a unique family \(Y_c\in \mathrm{GL}_{N_c}(\mathbb C)\) on the letter support:

The multiplicities in exact biorthogonal fusion and action decompositions give, at every positive chain length \(n\),

Both identities concern the original, unblocked tensors.

If positive-dimensional target blocks span their product matrix algebra simultaneously at length \(L\), their periodic block vectors are linearly independent at \(L\). On the doubled physical alphabet, the same conclusion gives independent periodic block operators.

Suppose the periodic fusion expansion in (26) holds and the operator blocks are separated, positive-dimensional, and injective. Then

Suppose the two periodic expansions in (26) hold with natural multiplicities and separated positive-dimensional injective state blocks. Then

For unweighted block assemblies of separated positive-dimensional injective operator and state blocks, arbitrary-boundary closedness and compatibility produce natural multiplicities \(N_{ab}^c\) and \(M_{ax}^y\) satisfying the periodic expansions (26), fusion associativity (27), and the NIM relation (28). No unit, duality, or commutativity assumption is needed.

Let \(B\) have two exact biorthogonal decompositions over a target family \((A_c)_c\), with repetitions indexed by maps \(f:I\to C\) and \(g:J\to C\). Suppose the simultaneous target word tuples span the product matrix algebra at one positive length. For every tuple of matrices \((M_c)_c\),

Suppose \(HS=I\), \(\widetilde H\widetilde S=I\), and \(SH=\widetilde S\widetilde H\). Then \(C=\widetilde HS\) and \(D=H\widetilde S\) obey

The common support need not be the ambient identity or self-adjoint.

Exact biorthogonal decompositions compose along finite two-stage paths: if the intermediate and final maps are \((V_i,W_i)\) and \((V'_{ij},W'_{ij})\), the composite maps are \(V'_{ij}V_i\) and \(W_iW'_{ij}\). Replacing \(B^a\) by \(PB^aQ\) with \(QP=I\) replaces its maps by \(V_iQ\) and \(PW_i\). Acting with a fixed MPO replaces state maps by \(I\otimes V_i\) and \(I\otimes W_i\); acting on a fixed state replaces operator maps by \(V_i\otimes I\) and \(W_i\otimes I\).

The collected maps obey \(HS=I\) and \(SH=\sum _iW_iV_i\). For two exact decompositions over target blocks spanning simultaneously at one positive word length, \(\widetilde HS\) and \(H\widetilde S\) are inverse full coordinate changes, and each carries the other synthesis matrix to its own.

Exact biorthogonal decomposition implies \(V_iB^a=A_i^aV_i\) and \(B^aW_i=W_iA_i^a\). Consequently \(V_i\widetilde W_j\) intertwines \(A_i\) with \(\widetilde A_j\). A bijective reindexing of the finite target copies preserves every decomposition identity.

If the pairwise fusion and action maps are exact and biorthogonal, both trees give exact biorthogonal decompositions of \((T_aT_b)\cdot A_x\). Their ambient support idempotents agree whenever the final state blocks have a simultaneous positive word span.

Under the exact-tree and spanning assumptions,

Assume the pairwise fusion and action maps are exact and biorthogonal. Suppose the state blocks are normal and positive-dimensional, and their word tuples span the product algebra at one length. In grouped final-block coordinates,

Assume exact biorthogonal fusion and action maps are given, and the state blocks are injective, positive-dimensional, and pairwise inequivalent under a virtual gauge and a nonzero scalar. The matrices constructed from these maps satisfy

Zero multiplicity spaces are allowed. In the source index order, (43) is the analysis-tree relation of [ 199 , Section 3 ] . Read the following trees from right to left: red legs have dimensions \(\chi _a,\chi _b,\chi _c\) and black legs have dimensions \(D_x,D_y,D_z\). The node indices are multiplicity labels, with row order \((z,i,j)\) and column order \((c,k,\mu )\). Both sides map \(\mathbb {C}^{\chi _a}\otimes \mathbb {C}^{\chi _b}\otimes \mathbb {C}^{D_x}\) to \(\mathbb {C}^{D_y}\), using the canonical identification of bracketings.

\(\displaystyle =\sum _{c,k,\mu }(L_{abx}^{y})_{c,k\mu }^{z,ij}\)

In prefix–suffix coordinates of lengths \(L,K\), every virtual boundary \(X\) satisfies

Empty prefixes and suffixes are allowed. If \(X\in \mathcal C(T)\), simultaneous cyclic rotation of two equal-length physical words leaves their boundary-weighted trace unchanged. Hence, for \(0\leq L\leq N\) and \(i\in \{ 0,\ldots ,N-1\} \), the same cut formula holds in the coordinates of the cyclic window starting at \(i\) and its ordered complement, with \(K=N-L\).

Let \(\mathcal B\) be a unital \(*\)-subalgebra of operators on the \(L\)-site physical space, and suppose that \(\mathcal S_A^L\) is invariant under \(\mathcal B\). Then \([h_L,B]=0\) for every \(B\in \mathcal B\). This statement also holds at \(L=0\).

Suppose \(T\) and \(A\) are compatible in the length-independent sense of Definition 30.1.2, and fix \(L{\gt}0\). Then \(\mathcal R_L(T)\) preserves \(\mathcal S_A^L\). If in addition \(\mathcal R_L(T)^*=\mathcal R_L(T)\), the unital \(*\)-algebra generated by this range preserves \(\mathcal S_A^L\), and \([h_L,O^L_{T,Y}]=[P_L,O^L_{T,Y}]=0\) for every virtual boundary \(Y\). Equivalently, \(h_L(O^L_{T,Y}v)=O^L_{T,Y}(h_Lv)\) for every physical vector \(v\), in either the configuration or orthonormal coordinates.

Suppose \(T\) and \(A\) are length-independently compatible, \(L_0{\gt}0\), and \(A\) is \(L_0\)-block injective. Let \(U\in M_{\chi }(\mathbb {C})\) satisfy \(O^{L_0}_{T,U}=I\). For every \(n{\gt}0\), put \(Q_n=O^n_{T,U}\). Then \(Q_n\psi ^n_{A,X}=\psi ^n_{A,X}\) for every \(X\in M_{D}(\mathbb {C})\), \(Q_nv=v\) for every \(v\in \mathcal S_A^n\), and \(Q_nP_n=P_n\). In particular, one-site injectivity and \(O^1_{T,U}=I\) suffice. There is no comparison between \(n\) and \(L_0\), and no adjoint-closure assumption is needed. This is a sufficient unit-support criterion for the normalization used in [ 199 , Appendix B ] .

Let \(0\leq L\leq N\) and \(i\in \{ 0,\ldots ,N-1\} \). Suppose an \(L\)-site operator \(B\) satisfies \([B,O^L_{T,Y}]=0\) for every virtual boundary \(Y\). Its placement \(B^{(i)}\) on the cyclic window starting at \(i\) satisfies \([B^{(i)},O^N_{T,X}]=0\) for every \(X\in \mathcal C(T)\).

Suppose \(T\) and \(A\) are compatible, \(0{\lt}L\leq N\), and \(\mathcal R_L(T)^*=\mathcal R_L(T)\). Then, for every cyclic placement \(i\) and every \(X\in \mathcal C(T)\), one has \([h_L^{(i)},O^N_{T,X}]=0\) and \([H_{L,N},O^N_{T,X}]=0\).

For \(T=\bigoplus _aT_a\), every word matrix is the direct sum of the component word matrices. Each \(Q_a\) commutes with every letter \(T^{ij}\), so \(Q_a\in \mathcal C(T)\), and \(O^N_{T,Q_a}=O^N_{T_a,I}\) for every \(N\geq 0\).

Let \(T=\bigoplus _a T_a\) be an unweighted finite direct sum compatible with \(A\). If \(0{\lt}L\leq N\) and \(\mathcal R_L(T)^*=\mathcal R_L(T)\), then \([H_{L,N},O^N_{T_a,I}]=0\) for every \(a\). Adjoint closure is required only of the entire assembled boundary range; taking adjoints may exchange labels.

For every \(L\geq 0\), the inequality \(d^L{\gt}D^2\) implies \(h_L\ne 0\). If also \(T\) and \(A\) are compatible, \(0{\lt}L\leq N\), and \(\mathcal R_L(T)^*=\mathcal R_L(T)\), this nonzero canonical interaction defines a periodic parent commuting with every \(O^N_{T,X}\) for \(X\in \mathcal C(T)\).

Suppose \(T\) and \(A\) are length-independently compatible, \(L_0,n{\gt}0\), \(A\) is \(L_0\)-block injective, and \(O^{L_0}_{T,U}=I\) for a fixed \(U\in M_{\chi }(\mathbb {C})\). Suppose also \(\mathcal R_n(T)^*=\mathcal R_n(T)\), and put \(Q_n=O^n_{T,U}\). Let \((X_j,Y_j)_{j\in J}\) be a finite family of pairs of virtual boundaries satisfying

Define \(E_n(B)=\sum _{j\in J}O^n_{T,X_j}BO^n_{T,Y_j}\). Then

The set \(J\) may be empty, and the physical and virtual dimensions may be zero, when the stated hypotheses hold. No identity \(Q_n=I\) or relation \(O^n_{T,Y_j}=(O^n_{T,X_j})^*\) is required. These are consequences of the product normalization in [ 199 , Appendix B ] for actual boundary operators.

Under the hypotheses of Theorem 30.4.11, if \(d^n{\gt}D^2\), then \(I-E_n(P_n)\ne 0\). This asserts no nonzeroness of \(E_n(h_n)\): if \(Q_n=P_n\), then \(E_n(h_n)=0\).

For every \(X\in M_{\chi }(\mathbb {C})\) and \(N\geq 0\),

Let \(P,Q\) have bond dimensions \(\chi ,\eta \), respectively, and let \(V:\mathbb {C}^\chi \to \mathbb {C}^\eta \), \(W:\mathbb {C}^\eta \to \mathbb {C}^\chi \) satisfy \(VW=I_\eta \), \(WV=I_\chi \), and \(VP^{ij}W=Q^{ij}\) for every \(i,j\). Then \(O^N_{P,X}=O^N_{Q,VXW}\) for every \(X\) and \(N\geq 0\).

Suppose (54) holds, \((a^\vee )^\vee =a\), and every \(T_a\), on its paired physical alphabet, is normal with \(\chi _a{\gt}0\). Then \(\chi _a=\chi _{a^\vee }\) for every \(a\). Here normality means that some positive blocking length makes the word matrices span \(M_{\chi _a}(\mathbb {C})\).

Under the hypotheses of Theorem 30.5.1.4, for each \(a\) there exist maps \(V_a:\mathbb {C}^{\chi _a}\to \mathbb {C}^{\chi _{a^\vee }}\) and \(W_a:\mathbb {C}^{\chi _{a^\vee }}\to \mathbb {C}^{\chi _a}\) with

The same maps satisfy, for every \(X\in M_{\chi _a}(\mathbb {C})\) and \(N\geq 0\),

No one-site injectivity, canonical normalization, or inequivalence between distinct blocks is required.

At every length \(N\geq 0\),

Suppose \(a\mapsto a^\vee \) is involutive, the \(T_a\) are normal with positive bond dimensions, and (54) holds. There exist maps \(V_a,W_a\) satisfying (55) and (56) for every \(a\), such that every \(X\) and \(N\geq 0\) satisfy

In particular the literal block sum is boundary adjoint closed.

Under the hypotheses of Theorem 30.5.2.5, let the literal block sum \(T\) be boundary compatible with a state tensor \(A\). Let \(P_{A,L}\) be the orthogonal projection onto its length-\(L\) arbitrary-boundary state space and let \(H_{A,L,N}\) be the periodic canonical parent formed by all cyclic translates of \(I-P_{A,L}\). For every \(0{\lt}L\leq N\) and every label \(a\),

No normality or injectivity of \(A\) is required.

Both \(T\), on its paired physical alphabet, and \(A\) are one-site injective, and hence normal.

The stacked operator tensor is exactly \(T\), and its action on the state tensor is exactly \(A\):

The tensor \(T\) is boundary closed and boundary compatible with \(A\). For all scalar boundaries \(x,y\) and every \(N{\gt}0\), the same output boundary \(xy\) satisfies

Every boundary commutes with every virtual tensor letter.

Already at length one,

Thus even the full one-site boundary range is not adjoint closed.

For \(f\in H^*\), define \(\kappa (f)(x)=\overline{f(x^*)}\). Then \(\kappa \) is conjugate-linear, involutive, and satisfies

Here multiplication is convolution, and \(1\) is the counit.

Suppose \(\phi (x^*)=\phi (x)^\dagger \). For every pair of physical configurations \(\sigma ,\tau \) of length \(N\geq 0\), put \(f_{\sigma ,\tau }=f_{\sigma _0\tau _0}\cdots f_{\sigma _{N-1}\tau _{N-1}}\), with the empty product equal to \(1\). Then

Suppose \(\phi \) preserves the involution and \(\psi \) is injective. For every \(X\in M_D(\mathbb C)\) there exists one \(Y\in M_D(\mathbb C)\) such that

for every \(N\geq 0\). In particular, \(T\) is boundary adjoint closed.

Under the hypotheses of Theorem 30.6.4, suppose \(T\) is arbitrary-boundary compatible with a state tensor \(A\). For \(L{\gt}0\), the canonical local parent commutes with \(O^L_{T,X}\) for every \(X\). For \(N\geq L\), the periodic canonical parent commutes with \(O^N_{T,X}\) whenever \(X\) commutes with every tensor letter \(T^{ij}\).

For operator tensors \(O,P\) and state tensors \(A,B\) on the same original physical alphabet, the products obey

Every equality retains the original ordered tensor product of virtual bond spaces. No positivity of physical or virtual dimensions is needed.

If the matrices \(O^{ij}\) span the full virtual matrix algebra, then the matrices \(E(O)^{ij}\) do so. If the matrices \(A^i\) span that algebra, then the matrices \(J(A)^{ij}\) do so.

Let \(c\) range over any finite set, possibly empty. Suppose rectangular matrices satisfy \(V_cW_c=\mathbb {1}\) and \(V_cW_b=0\) for \(c\ne b\). If \(O^{ij}=\sum _c W_cT_c^{ij}V_c\), then

If \(B^i=\sum _c W_cA_c^iV_c\), then

Thus both padded decompositions use the same target indices and the same analysis and synthesis matrices, with no ambient completeness requirement.

For equal virtual dimensions, a gauge-phase relation between \(E(O)\) and \(E(P)\) implies the same relation between \(O\) and \(P\). A gauge-phase relation between \(J(A)\) and \(J(B)\) implies the same relation between \(A\) and \(B\). In both cases, the invertible virtual matrix and nonzero scalar are unchanged.

Let the common virtual dimension be \(D{\gt}0\). If \(A\) is injective, no invertible \(X\) and nonzero scalar \(\zeta \) satisfy \(J(A)^{ij}=\zeta X E(O)^{ij}X^{-1}\) for every physical pair. If \(O\) is injective on its doubled physical alphabet, no such \(X,\zeta \) satisfy \(E(O)^{ij}=\zeta X J(A)^{ij}X^{-1}\) for every physical pair.

For a complete zipper fusion family, let \(A,B,C\) be the successive coordinate changes along the three-edge path from the fully left to the fully right associated fourfold fusion tree. Let \(D,E\) be the changes along the two-edge path, and let \(Q\) be the reverse of \(A\). Then

These identities hold on the finite multiplicity spaces, including when a multiplicity space is zero-dimensional.

Let \(N_{ab}^{c}\) be the multiplicities of a complete zipper fusion family, let \(P_{abc}^{g}\) be its forward triple comparison with right-tree rows and left-tree columns, and put \(Q_{abc}^{g}=(P_{abc}^{g})^{-1}\). Superscript tuples denote rows and subscript tuples denote columns. Fix labels \(a,b,c,d,e,f,g,j,i\) and multiplicity indices

Then

Here \(k\) runs over the block labels, while \(\tau ,\mu ,\nu ,\omega \) run respectively over the multiplicity spaces of dimensions \(N_{kj}^{e}\), \(N_{ab}^{k}\), \(N_{kc}^{g}\), and \(N_{fd}^{i}\). Empty multiplicity spaces contribute zero to these sums.

Suppose the original matrices are exact biorthogonal decompositions of every \(O_aO_b\) into the \(O_c\) channels and of every \(O_a\cdot A_x\) into the \(A_y\) channels. For every \(u,v\in \Lambda \), the matrices \(\widetilde V_{uv}^{w\mu },\widetilde W_{uv}^{w\mu }\) give an exact biorthogonal decomposition of \(T_uT_v\) into the \(T_w\) channels with multiplicities \(\widetilde N_{uv}^w\).

Let an operator family have any finite label set, and suppose a pairwise product has an exact biorthogonal decomposition into that family. Blocking the incoming tensors and every target tensor by a common positive length preserves the decomposition and both rectangular matrices.

Let \(T_c\) be a finite family of algebraically normal operator tensors with positive dimensions \(\Delta _c\). Suppose no two different labels are related by an invertible similarity and a nonzero scalar, and choose a bijection from the label set to a finite ordinal. Some common length \(L{\gt}0\) makes every \(T_c^{[L]}\) injective and admits coefficients \(K_{(c,u,v),(i,j)}\) satisfying

The right side is zero for different labels; for equal labels the virtual coordinates are compared in their common bond space.

Suppose every \(O_a\) on its doubled physical alphabet and every \(A_x\) is injective, with \(\chi _a,D_x{\gt}0\). Exclude scalar-gauge duplicates separately among the operators and among the states. Then the disjoint-union family has a common positive blocking length, individually injective blocked tensors, and a simultaneous inverse satisfying (86). No simultaneous inverse or joint spanning condition is a premise.

Use fusion analysis and synthesis \(W,\widehat W\) and action analysis and synthesis \(V,\widehat V\), as in Definition 30.3.7. Write \(u=((\alpha ,\beta ),\xi )\) for the common incoming bond, where \(\alpha {\lt}\chi _a\), \(\beta {\lt}\chi _b\), and \(\xi {\lt}D_x\), and let \(z{\lt}D_y\). For the paths \((c,\mu ;y,k)\) and \((t,j;y,i)\), respectively,

These formulas hold for arbitrary matrices of the indicated shapes. Empty intermediate bond spaces give empty sums.

Fix a complete zipper fusion family and final label \(d\) with bond dimension \(n_d{\gt}0\). Let \(P_{abc}^{d}\) be its forward comparison and \(Q_{abc}^{d}=(P_{abc}^{d})^{-1}\). For a left-tree multiplicity \(t\) and a right-tree multiplicity \(q\), let \(H_{\rm L}^{t},H_{\rm R}^{q}\) retain the final bond rows of the corresponding analysis maps, and let \(S_{\rm L}^{t},S_{\rm R}^{q}\) retain their synthesis columns. Then

Suppose the operator blocks have positive bond dimensions, are individually injective, admit the supplied exact fusion decomposition, and their simultaneous one-letter block map has a left inverse. The complete zipper fusion family constructed from these data satisfies

for every \(t\in \mathcal I_{\rm L}(a,b,c;d)\) and \(q\in \mathcal I_{\rm R}(a,b,c;d)\).

Label auxiliary operator blocks by \(o(a)\) and state blocks by \(q(x)\), with bond dimensions \(\chi _a\) and \(D_x\). Give operator/operator channels the original fusion maps and operator/state channels the original action maps; all other multiplicities are zero. Suppose an auxiliary tensor family with these maps has positive bonds, individual injectivity, exact fusion decompositions, and a simultaneous one-letter block left inverse. Its constructed forward comparison obeys

whenever \(i{\lt}M_{az}^{y}\), \(j{\lt}M_{bx}^{z}\), \(k{\lt}M_{cx}^{y}\), and \(\mu {\lt}N_{ab}^{c}\). Thus the auxiliary row \((z,j,i)\) is reordered as \((z,i,j)\) and the auxiliary column \((c,\mu ,k)\) as \((c,k,\mu )\).

Let an auxiliary tensor family on the disjoint operator/state labels have positive bond dimensions, individual injectivity, and a simultaneous one-letter block left inverse. Suppose the original fusion and action maps give its exact biorthogonal decompositions on the two populated kinds of channels, with every other multiplicity zero. Construct its complete zipper family from these data. On four operator labels, the inverse comparison is the source-oriented fusion matrix:

Here \(Q\) is the inverse of the forward comparison \(P\), with left-tree rows and right-tree columns. The auxiliary physical alphabet may already result from a common blocking; the coefficient depends only on the unchanged virtual maps.

Assume the auxiliary tensors, inverse, and exact maps satisfy the hypotheses of Theorem 30.11.1. Let \(a,b,c,e,f\) be operator labels and \(x,y,z,t\) state labels. Fix multiplicities

The normalized-trace matrices of the original maps satisfy

The left sum ranges over operator labels \(d\) and \(n\in \{ 0,\ldots ,M_{dt}^{y}-1\} \), \(\eta \in \{ 0,\ldots ,N_{ab}^{d}-1\} \), \(\chi \in \{ 0,\ldots ,N_{dc}^{e}-1\} \); the right sum ranges over \(l\in \{ 0,\ldots ,M_{fx}^{z}-1\} \). Empty multiplicity spaces are included.

Let the operator and state blocks be individually injective, have positive bond dimensions, and have no scalar-gauge duplicates within either family. Any exact biorthogonal fusion and action maps for these blocks satisfy (95) with their original unblocked normalized-trace coefficients.

Let \(T=\bigoplus _aO_a\) and \(B=\bigoplus _xA_x\) be the unweighted assembled operator and state tensors. Suppose every pair of operator boundaries \(X,Y\) admits a single boundary \(Z\) such that \(O^n_{T,X}O^n_{T,Y}=O^n_{T,Z}\) for every \(n{\gt}0\). Suppose also that every operator boundary \(X\) and state boundary \(Y\) admits a single state boundary \(Z\) such that \(O^n_{T,X}\psi ^n_{B,Y}=\psi ^n_{B,Z}\) for every \(n{\gt}0\). In both assumptions the output boundary is chosen before the positive length. Assume the individual blocks are injective, their virtual dimensions are positive, and no two distinct blocks within either original family are related by an invertible gauge and a nonzero scalar. Then there exist multiplicities \(N,M\) and exact biorthogonal fusion and action maps whose normalized-trace coefficients satisfy (95).

Let \(\{ T_g\} _{g\in G}\) be matrix product operator tensors of a group \(G\) whose doubled-index tensors are injective, with \(U_gU_h=U_{gh}\) and \(U_e=\mathbb {1}\) on every nonempty chain, and let \((F^{\lt}_{g,h},F^{\gt}_{g,h})\) be fusion tensors reducing \(T_gT_h\) onto \(T_{gh}\). Suppose the fusion is closed, \((T_gT_h)^{ij}=F^{\gt}_{g,h}\, T_{gh}^{ij}\, F^{\lt}_{g,h}\) for all \(g,h\) and all physical indices \(i,j\). Then every \(T_g\) has bond dimension one, the representation is on-site, \(U_g=u_g^{\otimes N}\) with \((u_g)_{ij}=T_g^{ij}\), and the anomaly three-cocycle \(\omega \) of the fusion tensors is a coboundary, so its class in \(H^{3}(G,\mathbb {C}^{\times })\) vanishes. The on-site conclusion does not use the product law.

For every label \(a\), the Perron–Frobenius dimension \(d_a\) is an eigenvalue of \(N_a\), and there is a column vector \(v\neq 0\) with nonnegative entries such that \(N_av=d_av\).

Let \(B\) be a matrix product state tensor and \(A\) a word-normal tensor of positive bond dimension, over the same alphabet. If a scalar \(c\in \mathbb {C}\) satisfies \(\operatorname{tr}(B^w)=c\, \operatorname{tr}(A^w)\) for every nonempty word \(w\), then \(c\) is a nonnegative integer.

Let \(T\) be a matrix product operator tensor and \(A\) a word-normal matrix product state tensor of positive bond dimension, with the same physical alphabet. Suppose there is a single scalar \(c\in \mathbb {C}\) such that, for every \(N\geq 1\),

Then \(c\) is a nonnegative integer.

Let \(\mu _1,\ldots ,\mu _n\) be nonzero complex numbers with \(\left|\sum _k\mu _k^L\right|=1\) for every \(L\geq 1\). Then \(n=1\) and \(|\mu _1|=1\).

Let \(T\) be a matrix product operator tensor, \(A\) a matrix product state tensor, and \(B\) an injective matrix product state tensor of positive bond dimension, all with the same physical alphabet. Suppose that for every \(N\geq 1\)

Then there is \(\lambda \in \mathbb {C}\) with \(|\lambda |=1\) and \(c_N=\lambda ^N\) for every \(N\geq 1\). In particular this holds when \(T\) is a matrix product unitary and \(A\) is injective, as in  [ 200 , Section II.B ] .

Let \((A_y)_{y\in K}\) be normal tensors of positive bond dimension with pairwise non-isomorphic word modules, let \(B\) be a tensor on the same physical alphabet, and let \(c_y\in \mathbb {C}\) satisfy \(\operatorname{tr}(B^w)=\sum _yc_y\operatorname{tr}(A_y^w)\) for every nonempty word \(w\). Then every \(c_y\) is a nonnegative integer.

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 ] .

If a normal tensor of positive bond dimension is symmetric under a fusion algebra with eigenvalues \(c_a\) and \(c_e=1\), then \(c_a\in \mathbb {N}\) for every \(a\) and \(a\mapsto c_a\) is a fusion character.

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 \(G\) be a finite group and \(X\) a finite set of block labels. Suppose the nonnegative integer multiplicities satisfy \(M_{gh,x}^y=\sum _z M_{h,x}^zM_{g,z}^y\) and \(M_{e,x}^y=\delta _{y,x}\). There is a unique action of \(G\) on \(X\) such that \(M_{g,x}^y=\delta _{y,g\cdot x}\). This is the permutation representation of [ 199 , Section 4 ] .

Let a finite group of periodic matrix product operators satisfy \(O_g^{(N)}O_h^{(N)}=O_{gh}^{(N)}\) for every \(N\geq 1\). Let \(A_x\) be a finite family of normal tensors of positive bond dimension whose periodic vectors are linearly independent at one positive length. Suppose that, for coefficients independent of \(N\),

and that \(O_e^{(N)}\) fixes every block vector. There is one group action on the block labels such that \(M_{g,x}^y=\delta _{y,g\cdot x}\) and, for every \(g,x\) and \(N\geq 1\),

Let \(M\) be a nonnegative integer representation of the Fibonacci ring on a set \(K\) of blocks with \(M_{1,x}^y=\delta _{xy}\). If \(K\) has one element, no such \(M\) exists. If \(K\) has two elements, then after labelling the blocks by \(1\) and \(\tau \) one has \(M_a=N_a\) for both labels, that is \(\tau \cdot x_1=x_\tau \) and \(\tau \cdot x_\tau =x_1+x_\tau \). Consequently a nonempty symmetric family of at most two normal tensors as in Theorem 30.15.8, on which the unit acts trivially, has exactly two blocks transforming in this way. The unrestricted direct-sum classification follows below; the unique indecomposable summand is this regular two-block representation [ 199 , Section 6 ] .

Let \(T\) be a matrix indexed by a finite set \(K\), with entries in \(\mathbb {N}\), satisfying \(T^2=I+T\). Set \(R=\{ x\in K:T_{xx}=0\} \) and \(F=\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}\). There is a bijection \(\sigma :K\longrightarrow R\times \{ 0,1\} \) for which

In particular, \(T\) is symmetric. Every unital nonnegative integer representation of the Fibonacci fusion ring is therefore a direct sum of regular representations. This gives the decomposition underlying the two-block action of [ 199 , Section 6 ] ; the correction adding the missing indecomposability hypothesis is recorded in [ 165 ] .

Let \(O_1,O_\tau \) be a Fibonacci matrix product operator algebra and \((A_x)_{x\in K}\) a finite family of normal tensors of positive bond dimensions. Suppose its periodic vectors are invariant with length-independent coefficients \(M_{a,x}^y\), and linearly independent at a positive length \(N_0\). Assume that \(O_1^{(N_0)}\) fixes each of those vectors. Set \(R=\{ x\in K:M_{\tau ,x}^x=0\} \). There is a bijection \(\sigma :K\longrightarrow R\times \{ 0,1\} \) such that

In particular, \(|K|=2|R|\) is even. This applies to arbitrary finite \(K\), including the empty family, and to the Fibonacci operators of Section 31.6 [ 199 , Sections 3 and 6 ] .

A nonempty finite indecomposable nonnegative integer representation of the Fibonacci ring, with the unit acting as the identity, has exactly two blocks. Here indecomposability means that \(M_\tau \) cannot be permuted into block-triangular form with two nonempty diagonal blocks. The blocks can be labelled \(x_1,x_\tau \) so that \(\tau \cdot x_1=x_\tau \) and \(\tau \cdot x_\tau =x_1+x_\tau \) [ 199 , Section 6 ] .

The label \(0\) is the unit of \(su(2)_4\) and \(1\) is the unit of \(\mathrm{Rep}(S_3)\). The assignment \(1,\pi ,\psi \mapsto 0,1,2\) identifies \(\mathrm{Rep}(S_3)\) with the fusion subring of integer labels of \(su(2)_4\): the fusion rules agree, and products of integer labels contain no half-integer label [ 199 , Section 6 ] .

The following actions on blocks, with the unit acting as the identity, satisfy (100).

1. For \(su(2)_4\): the fusion rules on five blocks, one per label; and the module \(\mathcal M_{TY}\) on blocks \(x,y,z,s\), where \(0\) and \(2\) fix every block, \(\tfrac 12\) and \(\tfrac 32\) send \(x,y,z\) to \(s\) and \(s\) to \(x+y+z\), and \(1\) sends \(x\) to \(y+z\), \(y\) to \(x+z\), \(z\) to \(x+y\) and \(s\) to \(2s\).

2. For \(\mathrm{Rep}(S_3)\): on one block \(s\), \(\pi \cdot s=2s\) and \(\psi \cdot s=s\) (the subgroup \(\mathbb {Z}_1\)); on blocks \(x,y\), \(\pi \cdot x=\pi \cdot y=x+y\) and \(\psi \) exchanging \(x\) and \(y\) (\(\mathbb {Z}_2\)); on blocks \(x,y,z\), \(\pi \cdot x=y+z\), \(\pi \cdot y=x+z\), \(\pi \cdot z=x+y\) and \(\psi \) fixing every block (\(\mathbb {Z}_3\)); and the fusion rules on three blocks (\(S_3\)).

These are the phases listed in [ 199 , Section 6 ] , except that the source prints the \(\mathbb {Z}_3\) action with \(\psi \) exchanging \(y\) and \(z\); that action does not satisfy (100), since \(\pi \cdot (\pi \cdot y)=x+2y+z\) while \((1+\pi +\psi )\cdot y=x+y+2z\) [ 161 ] . That the lists are complete is not treated here beyond one block (Theorem 30.15.25; [ 162 ] ).

Under the identification of \(\mathrm{Rep}(S_3)\) with the integer labels of \(su(2)_4\), the integer labels do not mix the blocks \(\{ x,y,z\} \) and \(\{ s\} \) of \(\mathcal M_{TY}\); they act on \(\{ x,y,z\} \) as in the \(\mathbb {Z}_3\) phase and on \(\{ s\} \) as in the \(\mathbb {Z}_1\) phase. They do not mix the blocks \(\tfrac 12,\tfrac 32\) of the first \(su(2)_4\) phase with the other blocks, and act on them as in the \(\mathbb {Z}_2\) phase [ 199 , Section 6 ] .

The only fusion character of \(\mathrm{Rep}(S_3)\) with values in \(\mathbb {N}\) is \((m_1,m_\pi ,m_\psi )=(1,2,1)\), the \(\mathbb {Z}_1\) phase; \(su(2)_4\) has no fusion character with values in \(\mathbb {N}\).

For a fusion ring there are positive numbers \(r_b\) with \(\sum _br_bN_{ab}^c=d_a\, r_c\) for all labels \(a\) and \(c\), where \(d_a\) is the Perron–Frobenius dimension of \(a\), and \(d_a{\gt}0\) for every \(a\).

Let \(N\) be a fusion ring with unit \(e\) and let \(M_{a,x}^y\) be a nonnegative integer representation of it on a finite set \(K\) with \(M_{e,x}^y=\delta _{xy}\). Then there are positive numbers \(v_x\) with

for all labels \(a\) and all \(y\in K\). If \(K\) is nonempty, \(d_a\) is the spectral radius of the matrix \(M_a\). No indecomposability of the representation and no commutativity of the fusion ring is needed.

Let \((A_x)_{x\in K}\), with \(K\) nonempty, 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 whose structure constants form a fusion ring with unit \(e\), and suppose \(M_{e,x}^y=\delta _{xy}\). Then there are positive numbers \(v_x\) such that for every label \(a\) and every \(N\geq 1\),

with \(d_a{\gt}0\), and \(|\psi ^{(N_0)}\rangle \neq 0\). This is the symmetric boundary state of [ 199 , Section 5 ] , whose eigenvalue \(r_a\) is the spectral radius of \(M_a\).

The unweighted letters span \(\operatorname{End}(V)\). The tensor \(A(\gamma )\) depends continuously on \(\gamma \in \mathbb {R}\) and is injective for \(0{\lt}\gamma {\lt}1\).

If \(B\) is injective and \(W\neq 0\), then \(\Gamma _{B,W}\) is injective and \(\dim \mathcal S'_{B,W}=D^2\). Invertibility of \(W\) is unnecessary.

The spaces \(\mathcal S'_\gamma \) have dimension \(D^2\) for every real \(\gamma \), and their orthogonal projectors are continuous. For \(0{\lt}\gamma {\lt}1\), the space \(\mathcal S'_\gamma \) equals the canonical two-site MPS support of \(A(\gamma )\). Consequently \(h_\gamma \) extends the canonical local parent interaction continuously to both endpoints. In particular, its interior limit at \(p=0,1\) is \(I-\Pi _{\mathcal S'_p}\).

For every inserted matrix \(W\) and boundary matrix \(X\),

Hence \(\mathcal S'_{A,W}\) is invariant under both outer sector projections.