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Lawrence–Krammer representation
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<h2 id="definition">Definition</h2> <p>Consider the <a href="/facts/Braid_group/rn5323Kx">braid group</a> B n {\displaystyle B_{n}} to be the <a href="/facts/Mapping_class_group/buQm001q">mapping class group</a> of a disc with <i>n</i> marked points, P n {\displaystyle P_{n}} . The Lawrence–Krammer representation is defined as the action of B n {\displaystyle B_{n}} on the homology of a certain <a href="/facts/Covering_map/bDRGasl7">covering</a> space of the <a href="/facts/Configuration_space_(physics)/LglVrWpq">configuration space</a> C 2 P n {\displaystyle C_{2}P_{n}} . Specifically, the first integral <a href="/facts/Homology_(mathematics)/VFJmryEW">homology group</a> of C 2 P n {\displaystyle C_{2}P_{n}} is isomorphic to Z n + 1 {\displaystyle \mathbb {Z} ^{n+1}} , and the subgroup of H 1 ( C 2 P n , Z ) {\displaystyle H_{1}(C_{2}P_{n},\mathbb {Z} )} invariant under the action of B n {\displaystyle B_{n}} is primitive, free abelian, and of rank 2. Generators for this invariant subgroup are denoted by q , t {\displaystyle q,t} . </p><p>The covering space of C 2 P n {\displaystyle C_{2}P_{n}} corresponding to the kernel of the projection map </p> π 1 ( C 2 P n ) → Z 2 ⟨ q , t ⟩ {\displaystyle \pi _{1}(C_{2}P_{n})\to \mathbb {Z} ^{2}\langle q,t\rangle } <p>is called the Lawrence–Krammer cover and is denoted C 2 P n ¯ {\displaystyle {\overline {C_{2}P_{n}}}} . <a href="/facts/Diffeomorphism/855N5YYj">Diffeomorphisms</a> of P n {\displaystyle P_{n}} act on P n {\displaystyle P_{n}} , thus also on C 2 P n {\displaystyle C_{2}P_{n}} , moreover they lift uniquely to diffeomorphisms of C 2 P n ¯ {\displaystyle {\overline {C_{2}P_{n}}}} which restrict to the identity on the co-dimension two boundary stratum (where both points are on the boundary circle). The action of B n {\displaystyle B_{n}} on </p> H 2 ( C 2 P n ¯ , Z ) , {\displaystyle H_{2}({\overline {C_{2}P_{n}}},\mathbb {Z} ),} <p>thought of as a </p> Z ⟨ t ± , q ± ⟩ {\displaystyle \mathbb {Z} \langle t^{\pm },q^{\pm }\rangle } -module, <p>is the Lawrence–Krammer representation. The group H 2 ( C 2 P n ¯ , Z ) {\displaystyle H_{2}({\overline {C_{2}P_{n}}},\mathbb {Z} )} is known to be a free Z ⟨ t ± , q ± ⟩ {\displaystyle \mathbb {Z} \langle t^{\pm },q^{\pm }\rangle } -module, of rank n ( n − 1 ) / 2 {\displaystyle n(n-1)/2} . </p> <h2 id="matrices">Matrices</h2> <p>Using Bigelow's conventions for the Lawrence–Krammer representation, generators for the group H 2 ( C 2 P n ¯ , Z ) {\displaystyle H_{2}({\overline {C_{2}P_{n}}},\mathbb {Z} )} are denoted v j , k {\displaystyle v_{j,k}} for 1 ≤ j < k ≤ n {\displaystyle 1\leq j<k\leq n} . Letting σ i {\displaystyle \sigma _{i}} denote the standard Artin generators of the <a href="/facts/Braid_group/rn5323Kx">braid group</a>, we obtain the expression: </p><p> σ i ⋅ v j , k = { v j , k i ∉ { j − 1 , j , k − 1 , k } , q v i , k + ( q 2 − q ) v i , j + ( 1 − q ) v j , k i = j − 1 v j + 1 , k i = j ≠ k − 1 , q v j , i + ( 1 − q ) v j , k − ( q 2 − q ) t v i , k i = k − 1 ≠ j , v j , k + 1 i = k , − t q 2 v j , k i = j = k − 1. {\displaystyle \sigma _{i}\cdot v_{j,k}=\left\{{\begin{array}{lr}v_{j,k}&i\notin \{j-1,j,k-1,k\},\\qv_{i,k}+(q^{2}-q)v_{i,j}+(1-q)v_{j,k}&i=j-1\\v_{j+1,k}&i=j\neq k-1,\\qv_{j,i}+(1-q)v_{j,k}-(q^{2}-q)tv_{i,k}&i=k-1\neq j,\\v_{j,k+1}&i=k,\\-tq^{2}v_{j,k}&i=j=k-1.\end{array}}\right.} </p> <h2 id="faithfulness">Faithfulness</h2> <p><a href="/facts/Stephen_Bigelow/GrIdCdGF">Stephen Bigelow</a> and Daan Krammer have given independent proofs that the Lawrence–Krammer representation is <a href="/facts/Group_representation/FsuuDagQ">faithful</a>. </p> <h2 id="geometry">Geometry</h2> <p>The Lawrence–Krammer representation preserves a non-degenerate <a href="/facts/Sesquilinear_form/Kp6qUA8v">sesquilinear form</a> which is known to be negative-definite Hermitian provided q , t {\displaystyle q,t} are specialized to suitable unit complex numbers (<i>q</i> near 1 and <i>t</i> near <i>i</i>). Thus the braid group is a subgroup of the <a href="/facts/Unitary_group/lEvudIhU">unitary group</a> of square matrices of size n ( n − 1 ) / 2 {\displaystyle n(n-1)/2} . Recently<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> it has been shown that the image of the Lawrence–Krammer representation is a <a href="/facts/Dense_set/nPT9Yxao">dense subgroup</a> of the <a href="/facts/Unitary_group/lEvudIhU">unitary group</a> in this case. </p><p>The sesquilinear form has the explicit description: </p><p> ⟨ v i , j , v k , l ⟩ = − ( 1 − t ) ( 1 + q t ) ( q − 1 ) 2 t − 2 q − 3 { − q 2 t 2 ( q − 1 ) i = k < j < l or i < k < j = l − ( q − 1 ) k = i < l < j or k < i < j = l t ( q − 1 ) i < j = k < l q 2 t ( q − 1 ) k < l = i < j − t ( q − 1 ) 2 ( 1 + q t ) i < k < j < l ( q − 1 ) 2 ( 1 + q t ) k < i < l < j ( 1 − q t ) ( 1 + q 2 t ) k = i , j = l 0 otherwise {\displaystyle \langle v_{i,j},v_{k,l}\rangle =-(1-t)(1+qt)(q-1)^{2}t^{-2}q^{-3}\left\{{\begin{array}{lr}-q^{2}t^{2}(q-1)&i=k<j<l{\text{ or }}i<k<j=l\\-(q-1)&k=i<l<j{\text{ or }}k<i<j=l\\t(q-1)&i<j=k<l\\q^{2}t(q-1)&k<l=i<j\\-t(q-1)^{2}(1+qt)&i<k<j<l\\(q-1)^{2}(1+qt)&k<i<l<j\\(1-qt)(1+q^{2}t)&k=i,j=l\\0&{\text{otherwise}}\\\end{array}}\right.} </p> <h2 id="further-reading">Further reading</h2> <ul><li><a href="/facts/Stephen_Bigelow/GrIdCdGF">Bigelow, Stephen</a> (2001), "Braid groups are linear", <i><a href="/facts/Journal_of_the_American_Mathematical_Society/qR98drCm">Journal of the American Mathematical Society</a></i>, 14 (2): 471–486, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0894-0347-00-00361-1">10.1090/S0894-0347-00-00361-1</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1815219">1815219</a></li> <li><a href="/facts/Stephen_Bigelow/GrIdCdGF">Bigelow, Stephen</a> (2003), "The Lawrence–Krammer representation", <i>Topology and geometry of manifolds</i>, Proceedings of Symposia in Pure Mathematics, vol. 71, Providence, RI: American Mathematical Society, pp. 51–68, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2Fpspum%2F071">10.1090/pspum/071</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780821835074, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2024629">2024629</a></li> <li>Budney, Ryan (2005), "On the image of the Lawrence–Krammer representation", <i><a href="/facts/Journal_of_Knot_Theory_and_Its_Ramifications/R0wydByd">Journal of Knot Theory and Its Ramifications</a></i>, 14 (6): 773–789, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0202246">math/0202246</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1142%2FS0218216505004044">10.1142/S0218216505004044</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2172897">2172897</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:14196563">14196563</a></li> <li>Krammer, Daan (2002), "Braid groups are linear", <i><a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a></i>, 155 (1): 131–156, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0405198">math/0405198</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F3062152">10.2307/3062152</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/3062152">3062152</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1888796">1888796</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:62899383">62899383</a></li> <li><a href="/facts/Ruth_Lawrence/onYL6Ru9">Lawrence, Ruth</a> (1990), <a href="http://projecteuclid.org/euclid.cmp/1104201923">"Homological representations of the Hecke algebra"</a>, <i><a href="/facts/Communications_in_Mathematical_Physics/Akpn9mU1">Communications in Mathematical Physics</a></i>, 135 (1): 141–191, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1990CMaPh.135..141L">1990CMaPh.135..141L</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf02097660">10.1007/bf02097660</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1086755">1086755</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:121644260">121644260</a></li> <li>Paoluzzi, Luisa; Paris, Luis (2002). "A note on the Lawrence–Krammer–Bigelow representation". <i><a href="/facts/Algebraic_and_Geometric_Topology/Mgmju995">Algebraic and Geometric Topology</a></i>. 2: 499–518. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0111186">math/0111186</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2140%2Fagt.2002.2.499">10.2140/agt.2002.2.499</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1917064">1917064</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:12672756">12672756</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Bigelow, Stephen (2003), "The Lawrence–Krammer representation", Topology and geometry of manifolds, Proc. Sympos. Pure Math., vol. 71, Providence, RI: Amer. Math. Soc., pp. 51–68, MR 2024629 <a href="/wiki/Stephen_Bigelow" target="_blank">/wiki/Stephen_Bigelow</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Budney, Ryan (2005), "On the image of the Lawrence–Krammer representation", Journal of Knot Theory and Its Ramifications, 14 (6): 773–789, arXiv:math/0202246, doi:10.1142/S0218216505004044, MR 2172897, S2CID 14196563 <a href="/wiki/Journal_of_Knot_Theory_and_Its_Ramifications" target="_blank">/wiki/Journal_of_Knot_Theory_and_Its_Ramifications</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>