{"id":"8f919cec-91d4-412f-9a8d-c93ed21ff034","arxiv_id":"2506.03806","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Every complex local representation of TVB2 into GL3(C) belongs to one of eight explicit families, and similar families are listed for TVBn into GL_{n+1}(C) and for STVB2 into M3(C).","lead":"This paper classifies all complex local representations of the twisted virtual braid group TVB2 into 3x3 matrices, listing eight types, and gives analogous classifications for larger twisted and singular-twisted virtual braid groups. Generalists might care because these classifications are a step toward deciding whether these newer braid-like groups are linear, a question with roots in knot theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.3's reducibility claims are false: an allowed ζ2 parameter choice reduces to degree 1, and ζ3/ζ4 always reduce because ±I2 blocks have every vector as an eigenvector.","rationale":"Good-faith reading: the paper's core contribution is a classification of local representations (Theorems 3.1, 3.4, 4.1) together with faithfulness and reducibility statements. The proof of the n=2 classification is compressed ('solving this system') and the n≥3 and STVB2 classifications are delegated to 'a similar proof', which alone would justify caution. But there is a sharper, decisive problem: Theorem 3.3 is not merely under-proved; it is false at an allowed parameter point. The ζ2 example satisfies the hypotheses of Theorem 3.1 (x≠0 and the non-degeneracy condition) and has a one-dimensional invariant subspace, so the blanket claim in 3.3(i) cannot stand. Likewise, ζ3 and ζ4 always admit a common eigenvector because the relevant ρ and γ blocks are ±I2 and I2; the proof's coordinate-vector assumption is invalid. Since the abstract explicitly advertises the reducibility analysis, and since Theorem 4.3 relies on the same argument, the paper's presented results are internally inconsistent. This reinforces the reader's REJECT verdict rather than changing it; the recommended verdict is therefore UNCHANGED. The proposed concrete test makes the counterexample fully explicit and mechanical. If a re-derivation were somehow to exclude the parameter point, the burden would shift to the authors to state the correct parameter restrictions and revisit the irreducibility classification.","tokens_in":18380,"tokens_out":26384,"duration_ms":256501,"concrete_test":"Directly verify the ζ2 instance b=1, d=0, w=0, x=1 (x≠0 and the stated non-degeneracy condition hold): confirm ζ2(σ1)v=ζ2(ρ1)v=v for v=(1,1,0) and γ1=γ2=I3. If so, span(v) is a one-dimensional invariant subspace and Theorem 3.3(i) is false. Then check ζ4 with S=[[1,1],[0,1]]: since ρ1=γ1=γ2=I3, C(1,0,0) is invariant despite b≠0, refuting the 'only if' direction of Theorem 3.3(ii).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central classification of normal forms may be salvageable, but the advertised reducibility analysis is demonstrably wrong. Theorem 3.3(i) asserts that type ζ2 is never further reducible to degree 1. Take the allowed ζ2 parameters b=1, d=0, w=0, x=1. Then ζ2(σ1)=ζ2(ρ1)=[[0,1,0],[1,0,0],[0,0,1]] and ζ2(γ1)=ζ2(γ2)=I3. The line spanned by v=(1,1,0) is invariant: σ1v=ρ1v=v and γ1v=γ2v=v. Hence ζ2 is reducible to degree 1, contradicting 3.3(i). More systematically, in types ζ3 and ζ4 the relevant 2×2 blocks of ρ1 (and γ1,γ2) are ±I2 and I2, so every vector is an eigenvector; choosing any eigenvector of the σ1-block gives a common invariant line. Thus the condition in 3.3(ii), 'b=0 and a≠d', is not necessary. For example, ζ4 with S=[[1,1],[0,1]] has invariant line C(1,0,0) although b≠0. The proof's mistake is to assume the eigenvectors of γ or ρ must be the coordinate vectors (0,1) and (1,0); for ±I2 blocks all vectors qualify. Theorem 4.3 is proved by 'a similar proof' and inherits the same defect, and the abstract advertises this reducibility analysis as part of the results. The false statements are therefore load-bearing, not cosmetic.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper classifies complex local representations of the twisted virtual braid group TVB2 into GL3(C), obtaining eight normal forms ζ1,...,ζ8; homogeneous local representations of TVBn into GL_{n+1}(C) for n≥3, obtaining seven normal forms; and local representations of the singular twisted virtual braid group STVB2 into M3(C), obtaining thirteen normal forms. It analyzes faithfulness and irreducibility of these representations and compares local extensions with Φ-type extensions of the first family ζ1 to STVB2.","tokens_in":18719,"tokens_out":18272,"duration_ms":164288,"significance":"If the classification theorems are correct, the paper provides complete parameter lists for local representations in these settings, which would be a useful first step toward linearity and extension problems for twisted virtual braid groups. The families are given explicitly and are concrete enough to check. However, the irreducibility analysis contains demonstrable errors and the classification proofs omit essential computations, so the paper's advertised results are not established in the submitted form.","major_comments":[{"comment":"The claim that type ζ2 is never further reducible to degree 1 is false. Take b=0, d=2, w=0, x=1; these parameters satisfy the stated conditions (d≠0, x≠0). Then the 2×2 block of ζ2(σ1) is 2I2 and the 2×2 block of ζ2(ρ1) is the swap matrix [[0,1],[1,0]], while γ1 and γ2 are the identity. The vector (1,1,0) is a common eigenvector of all three matrices, so the representation is reducible to degree 1. The same conclusion follows with b=1, d=0, w=0, x=1, which also satisfies the stated conditions. The proof in §3 incorrectly restricts the search for common eigenvectors to the coordinate vectors (0,1) and (1,0).","section":"§3, Theorem 3.3(i)"},{"comment":"The criterion 'reducible to degree 1 if and only if b=0 and a≠d' for types ζ3 and ζ4 is wrong. In ζ3 the 2×2 block of ρ1 is −I2 and in ζ4 it is I2; in both cases the γ blocks are I2, so every vector is an eigenvector of these blocks. Hence any eigenvector of the 2×2 block of σ1 gives a common invariant line, and such an eigenvector always exists over C. For example, ζ4 with a=1, b=1, c=0, d=1 has σ1-block [[1,1],[0,1]], whose eigenvector (1,0) is also an eigenvector of I2 and of the γ blocks; thus the invariant line is C(1,0,0). The stated condition is therefore neither necessary nor sufficient.","section":"§3, Theorem 3.3(ii)"},{"comment":"Theorem 4.3 is proved by 'a similar proof as the proof of Theorem 3.3' and inherits the same defective eigenvector analysis. A concrete counterexample to part (i) is the family η2 with x=1, z=0, a=2, b=1, f=1, g=1. These parameters satisfy the condition a^2x^2+2abxz−b^2+b^2z^2=3≠0 and x≠0. The 2×2 blocks of σ1, ρ1, τ1 are respectively [[2,1],[1,2]], [[0,1],[1,0]], and [[1,1],[1,1]], all of which fix the vector (1,1) up to scale; γ1 and γ2 are the identity. Thus span{(1,1,0)} is invariant and the representation is further reducible to degree 1, contradicting the assertion that η1,...,η4 are not further reducible.","section":"§4, Theorem 4.3"},{"comment":"The completeness of the classifications is not verifiable from the submitted text. In the proof of Theorem 3.1, the reduction to q=r=0, s=1 is asserted after 'Solving this system gives directly' without presenting the 31-equation system or its solution, and the subsequent case analysis relies on this unreported step. The proofs of Theorem 3.4 and Theorem 4.1 say only 'a similar proof as the proof of Theorem 3.1 gives the required result' and provide no equations at all. Since the main contribution is a complete classification of all local representations in each setting, these omitted computations are load-bearing; the reader cannot rule out missing families or extra parameter restrictions.","section":"§3.1, §3.3, §4.1"}],"minor_comments":[{"comment":"The condition for ζ2, '(d ≠ 0 or dx ≠ ±b − bw)', is ambiguous; it should clarify that the disjunction is over d≠0 and dx ≠ ±(b−bw), and the parenthesization should be fixed.","section":"§3, Theorem 3.1"},{"comment":"In Case 3 of the proof, the text refers to 'a > d' and 'd > a' for complex numbers a,d; complex numbers are not ordered. The intended condition is a≠d, and the inequalities should be removed.","section":"§3, Theorem 3.3 proof"},{"comment":"In part (6), the parameter list says 'where b ∈ C∗' but the displayed matrices use the parameter x; this is a typo and should read x∈C∗. Also, the 2×2 blocks in parts (1)–(4) use expressions such as √b/√c and √c/√b, which require a choice of square-root branches for b,c∈C∗; the notation should be clarified or the families should be parameterized without roots.","section":"§3, Theorem 3.4"},{"comment":"The formulas for t, u, v have denominators involving b^3−b(d−1)^2x^2; the parameter conditions under which these denominators are nonzero are not stated.","section":"§5, Corollary 5.2"}],"recommendation":"major_revision","confidential_remarks":"The paper contains several demonstrable false statements in Theorems 3.3 and 4.3, but the classification part may be salvageable if the missing computations are supplied and the irreducibility analysis is redone. The authors should also re-examine the typesetting of the γ_j blocks in Theorem 3.4, which appear to be singular as printed. I would not accept the paper in its present form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The classification lists are the real contribution, and the reducibility analysis is wrong. Theorem 3.3(i) fails for ζ2: with b=1, d=0, w=0, x=1, both σ1 and ρ1 act as the swap on the first two coordinates and γ1=γ2=I3, so the line spanned by (1,1,0) is invariant—ζ2 is reducible to degree 1. The proof assumes eigenvectors of the γ1/γ2 blocks must be coordinate vectors; when those blocks are ±I2, every vector is an eigenvector. The same error breaks the 'if and only if' in 3.3(ii) and, by inheritance, Theorem 4.3. Since the abstract advertises this reducibility analysis, the false statements are not cosmetic.\n\nWhat is genuinely new: the paper gives the first classification of complex local representations of TVB2 into GL3(C), homogeneous local representations of TVBn into GL_{n+1}(C) for n≥3, and local representations of STVB2 into M3(C). The eight, seven, and thirteen families are new, and the n=2 derivation from the defining relations is mostly transparent. The unfaithfulness arguments are correct.\n\nThe soft spots are serious. The n≥3 and singular classifications are not actually proved: Theorem 3.4 and Theorem 4.1 say 'a similar proof' with no equations, so completeness cannot be checked. The citation pattern is fine; using [9] for the Bn irreducibility criterion is standard and not circular.\n\nBottom line: if the reducibility statements were corrected and the missing computations supplied, this would be a useful specialized paper for people working on braid group representations and linearity of twisted virtual braid groups. In its current form, it contains provably false theorems, so I would not cite it or accept it as is. A serious referee should still engage with it, because the classification is new and the fix is plausibly within reach.","headline":"New classification lists for twisted virtual braid group representations, but the reducibility theorems are demonstrably false and the n≥3 proofs are omitted; needs serious revision before it can be trusted.","tokens_in":19267,"tokens_out":6692,"would_cite":false,"duration_ms":65317,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F36","20F38","57K12"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper classifies every complex local representation of the twisted virtual braid group TVB2 into GL3(C) as one of eight explicit families, all unfaithful and reducible to 2x2, then extends the list to seven homogeneous families for…","keywords":["twisted virtual braid group","local representations","homogeneous local representations","singular twisted virtual braid monoid","faithfulness","irreducibility","Phi-type extensions","matrix representations"],"falsifier":"A direct computer-algebra check would settle the classification: generate the full solution set of the defining relations for $TVB_2$ with arbitrary invertible complex $2 \\times 2$ blocks for $\\gamma_1,\\gamma_2$, and compare the equivalence classes with the eight families of Theorem 3.1; any class not equivalent to a $\\zeta_i$ refutes completeness. For the reducibility statements, take $\\zeta_3$ or $\\zeta_4$ with the $2 \\times 2$ block of $\\gamma_1=\\gamma_2$ equal to $-I_2$ and a nonzero off-diagonal $\\sigma_1$ block, then test whether the $\\sigma_1$-block eigenvectors are common to all matrices; if such a common eigenvector exists outside the stated conditions, the 'only if' part of Theorem 3.3(iii) fails.","tokens_in":18154,"feed_emoji":"🧵","tokens_out":15651,"duration_ms":151566,"temperature":0.7,"pith_summary":"The paper sets out to classify all complex local representations of the twisted virtual braid group on two strands, $TVB_2$, into $\\mathrm{GL}_3(\\mathbb{C})$. Its main result is a complete list of eight explicit matrix families, $\\zeta_1$ through $\\zeta_8$, and it proves that every such representation is equivalent to one of them, that none is faithful, and that each reduces to a $2 \\times 2$ representation. The same programme is pushed further: for $n \\geq 3$, complex homogeneous local representations of $TVB_n$ into $\\mathrm{GL}_{n+1}(\\mathbb{C})$ are classified into seven unfaithful families, and complex local representations of the singular twisted virtual braid monoid $STVB_2$ into $\\mathrm{M}_3(\\mathbb{C})$ are classified into thirteen unfaithful families. In the last section the paper compares two natural ways to extend a $TVB_2$ representation to $STVB_2$, local extension and $\\Phi$-type extension, and shows that they are not the same mechanism. A complete catalogue of this kind matters because it is the first concrete step toward deciding whether these generalized braid groups admit any faithful linear representation at all.","feed_headline":"All 3x3 local representations of TVB2 fit eight families","feed_subtitle":"Complete classification shows all are unfaithful and reducible to 2x2, with analogs for larger n and singular twists.","key_machinery":"The engine of the classification is the local representation ansatz itself: each generator is represented by a block-diagonal matrix carrying one small block, a $2 \\times 2$ matrix for $TVB_2 \\to \\mathrm{GL}_3(\\mathbb{C})$, and identity blocks elsewhere, with the twist generators $\\gamma_j$ forming a family of length $n$. Plugging this ansatz into the defining relations of $TVB_n$, the involutions $\\rho_i^2=1$, $\\gamma_j^2=1$, the commutation rules, and the key mixed relation $\\rho_1\\sigma_1\\rho_1=\\gamma_2\\gamma_1\\sigma_1\\gamma_1\\gamma_2$, turns the representation problem into an explicit polynomial system; for $n=2$ it is a system of 31 equations in the 12 entries of the four $2 \\times 2$ blocks, whose solution and subsequent change-of-basis normalization produce the eight normal forms. For $n \\geq 3$ the same computation is run with all blocks constant, the homogeneous case, and for $STVB_2$ the additional singular relations $\\sigma_1\\tau_1=\\tau_1\\sigma_1$ and $\\rho_1\\tau_1\\rho_1=\\gamma_2\\gamma_1\\tau_1\\gamma_1\\gamma_2$ are imposed. The non-further-reducibility of the $n$-dimensional restrictions is controlled by restricting to the braid subgroup $B_n$ and invoking the lemma that irreducibility on a subgroup forces irreducibility of the ambient representation. The paper then separately promotes the classical $\\Phi$-type extension formula $\\Phi_{t,u,v}(\\tau_1)=t\\,\\zeta(\\sigma_1)+u\\,\\zeta(\\sigma_1^{-1})+vI_3$ to the twisted setting and compares it with the local extension.","core_discovery":"The paper's central claim is that the local ansatz is rigid for $TVB_2$: if $\\zeta \\colon TVB_2 \\to \\mathrm{GL}_3(\\mathbb{C})$ is any complex local representation, then after a change of basis its values are one of the eight families $\\zeta_1,\\dots,\\zeta_8$ displayed in Theorem 3.1, with parameters constrained by inequalities such as $b^2-d^2x^2 \\neq 0$ and $x \\neq 0$. In all eight families the twist generators $\\gamma_1,\\gamma_2$ act in only two ways, either both are the identity or they are $\\mathrm{diag}(-1,1,1)$ and $\\mathrm{diag}(1,-1,1)$, so the classification reduces the representation theory of this local type to a short list of $2 \\times 2$ blocks. Each family is unfaithful, and each contains the invariant line spanned by the third standard basis vector, hence each is reducible to a $2 \\times 2$ subrepresentation; Theorem 3.3 then states which families reduce further to degree 1. For $n \\geq 3$, Theorem 3.4 gives the analogous seven homogeneous local families in $\\mathrm{GL}_{n+1}(\\mathbb{C})$, and Theorem 4.1 gives thirteen families for $STVB_2$ into $\\mathrm{M}_3(\\mathbb{C})$. Finally, Theorem 5.1 and Corollary 5.2 show that the local extension $\\eta_1$ of $\\zeta_1$ agrees with a $\\Phi$-type extension only for a specific parameter formula, so not every local representation of $TVB_2$ extends to $STVB_2$ by the $\\Phi$-type prescription.","pith_inferences":["Because every classified local representation is unfaithful, any faithful complex representation of $TVB_n$ must abandon the local block form or use a dimension larger than those covered here; the smallest faithful degree of $TVB_n$ is an immediate open problem.","The equation-solving method is portable: repeating the classification over a finite field would turn parameter inequalities such as $b^2-d^2x^2 \\neq 0$ into congruences and could produce new normal forms when those expressions vanish.","The $\\Phi$-type comparison for $\\zeta_1$ likely represents a pattern: for each $\\zeta_i$, the locus of parameters where a local extension agrees with a $\\Phi$-type extension should be a proper algebraic subvariety, with Corollary 5.2 as the first explicit example.","A testable repair of the reducibility analysis is to compute joint eigenvectors of the $2 \\times 2$ blocks without assuming they are coordinate vectors; cases where the $\\gamma$-block is $-I_2$ admit extra candidate common eigenvectors and would shift the reducibility-to-1 criteria."],"forward_implications":["Every complex local representation of $TVB_2$ in $\\mathrm{GL}_3(\\mathbb{C})$ is accounted for by eight parameterized families; no such representation is faithful, so the twisted virtual braid group does not admit a faithful $3 \\times 3$ local representation.","All eight families reduce to a $2 \\times 2$ subrepresentation; types $\\zeta_3$ through $\\zeta_8$ sometimes reduce further to degree 1, with explicit conditions such as $b=0$ and $a \\neq d$ for $\\zeta_3$ and $\\zeta_4$.","For $n \\geq 3$, homogeneous local representations of $TVB_n$ into $\\mathrm{GL}_{n+1}(\\mathbb{C})$ are exhausted by seven families; for $\\zeta'_1$ through $\\zeta'_4$ the $n$-dimensional reduction stays irreducible when $bc \\neq 1$.","The singular twisted virtual braid monoid $STVB_2$ has thirteen local $\\mathrm{M}_3(\\mathbb{C})$ families, all unfaithful and reducible to $2 \\times 2$; when $\\eta(\\tau_1)$ is invertible these extend to $STVG_2$.","Local extension and $\\Phi$-type extension are different mechanisms: for $\\zeta_1$ they agree only when $t,u,v$ satisfy the three formulas in Corollary 5.2, so there are local extensions that are not $\\Phi$-type."],"supporting_citations":[{"why":"It introduces the twisted virtual braid group TVB_n and its defining relations, the object whose local representations are classified.","marker":"[23]"},{"why":"It introduces the singular twisted virtual braid monoid STVB_n and its Phi-type extension, which Section 5 compares against local extensions.","marker":"[25]"},{"why":"It defines Phi-type extensions of braid group representations to the singular braid monoid, the construction that Theorem 5.1 transplants to the twisted setting.","marker":"[3]"},{"why":"It supplies the local representation ansatz and the classification strategy for low-degree braid group representations that this paper extends.","marker":"[17]"},{"why":"It classifies homogeneous 3-local representations of the braid group, the result that Theorem 3.4 generalizes to TVB_n.","marker":"[19]"},{"why":"It provides the irreducibility criterion for local braid group representations that Theorem 3.7 uses to rule out further reducibility when bc is not 1.","marker":"[9]"},{"why":"It establishes the comparison between local extensions and Phi-type extensions for the singular braid monoid, the model for Corollary 5.2.","marker":"[21]"},{"why":"It supplies the standard definitions of representation, faithfulness, and irreducibility on which every statement in the paper depends.","marker":"[26]"}],"fun_headline_variants":["TVB2 local reps: eight unfaithful families, all reducible","Eight families classify all 3x3 local twisted virtual braid reps","All local TVB2 reps are unfaithful and reducible: eight types","Twisted virtual braid local reps classified: eight unfaithful families","Eight unfaithful families: complete classification of TVB2 local reps"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing step is an omitted calculation: for $n=2$ the paper states the solution of a 31-equation system in 12 unknowns without showing the full derivation, and for $n \\geq 3$ and for $STVB_2$ it asserts 'a similar proof' with no equations displayed, so the completeness of the seven- and thirteen-family lists rests entirely on those hidden computations. The reducibility claims additionally assume that any common eigenvector of the $2 \\times 2$ blocks must be a standard coordinate vector, which is not generally true when a block is $\\pm I_2$.","fun_headline_variants_meta":{"raw":{"variants":["TVB2 local reps: eight unfaithful families, all reducible","Eight families classify all 3x3 local twisted virtual braid reps","All local TVB2 reps are unfaithful and reducible: eight types","Twisted virtual braid local reps classified: eight unfaithful families","Eight unfaithful families: complete classification of TVB2 local reps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000765,"raw_usage":{"total_tokens":3466,"prompt_tokens":1089,"completion_tokens":2377,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":705,"completion_tokens_details":{"reasoning_tokens":2276}},"tokens_in":705,"tokens_out":2377,"duration_ms":20897,"temperature":1.0,"reasoning_tokens":2276,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:56:48.032407+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct computer-algebra check would settle the classification: generate the full solution set of the defining relations for $TVB_2$ with arbitrary invertible complex $2 \\times 2$ blocks for $\\gamma_1,\\gamma_2$, and compare the equivalence classes with the eight families of Theorem 3.1; any class not equivalent to a $\\zeta_i$ refutes completeness. For the reducibility statements, take $\\zeta_3$ or $\\zeta_4$ with the $2 \\times 2$ block of $\\gamma_1=\\gamma_2$ equal to $-I_2$ and a nonzero off-diagonal $\\sigma_1$ block, then test whether the $\\sigma_1$-block eigenvectors are common to all matrices; if such a common eigenvector exists outside the stated conditions, the 'only if' part of Theorem 3.3(iii) fails.","supporting_citations":[{"cited_title":"Twisted virtual braids and twisted links","cited_arxiv_id":"2302.13244","evidence_quote":"It introduces the twisted virtual braid group TVB_n and its defining relations, the object whose local representations are classified."},{"cited_title":"The monoid structure of singular twisted virtual braids","cited_arxiv_id":"2502.08965","evidence_quote":"It introduces the singular twisted virtual braid monoid STVB_n and its Phi-type extension, which Section 5 compares against local extensions."},{"cited_title":"Bardakov, N","cited_arxiv_id":null,"evidence_quote":"It defines Phi-type extensions of braid group representations to the singular braid monoid, the construction that Theorem 5.1 transplants to the twisted setting."},{"cited_title":"Mikhalchishina, Local Representations of Braid Groups, Sib","cited_arxiv_id":null,"evidence_quote":"It supplies the local representation ansatz and the classification strategy for low-degree braid group representations that this paper extends."},{"cited_title":"Chreif and M","cited_arxiv_id":null,"evidence_quote":"It provides the irreducibility criterion for local braid group representations that Theorem 3.7 uses to rule out further reducibility when bc is not 1."},{"cited_title":"Nasser, Local Extensions and Φ-Type Extensions of Some Local Representations of the Braid Group Bn to the Singular Braid Monoid SMn, Vietnam Journal of Mathematics, (2024) 1–12","cited_arxiv_id":null,"evidence_quote":"It establishes the comparison between local extensions and Phi-type extensions for the singular braid monoid, the model for Corollary 5.2."},{"cited_title":"Vinberg, Linear Representations of Groups, Translated from the Russian by A","cited_arxiv_id":null,"evidence_quote":"It supplies the standard definitions of representation, faithfulness, and irreducibility on which every statement in the paper depends."}],"review_version":1}