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Khovanov-Rozansky homology of Coxeter knots and Schröder polynomials for paths under any line 科克斯特结的 Khovanov-Rozansky 同调和任意线下路径的 Schröder 多项式
arXiv - MATH - Quantum Algebra Pub Date : 2024-07-25 DOI: arxiv-2407.18123
Carmen Caprau, Nicolle González, Matthew Hogancamp, Mikhail Mazin
{"title":"Khovanov-Rozansky homology of Coxeter knots and Schröder polynomials for paths under any line","authors":"Carmen Caprau, Nicolle González, Matthew Hogancamp, Mikhail Mazin","doi":"arxiv-2407.18123","DOIUrl":"https://doi.org/arxiv-2407.18123","url":null,"abstract":"We introduce a family of generalized Schr\"oder polynomials $S_tau(q,t,a)$,\u0000indexed by triangular partitions $tau$ and prove that $S_tau(q,t,a)$ agrees\u0000with the Poincar'e series of the triply graded Khovanov-Rozansky homology of\u0000the Coxeter knot $K_tau$ associated to $tau$. For all integers $m,n,dgeq 1$\u0000with $m,n$ relatively prime, the $(d,mnd+1)$-cable of the torus knot $T(m,n)$\u0000appears as a special case. It is known that these knots are algebraic, and as a\u0000result we obtain a proof of the $q=1$ specialization of the\u0000Oblomkov-Rasmussen-Shende conjecture for these knots. Finally, we show that our\u0000Schr\"oder polynomial computes the hook components in the Schur expansion of\u0000the symmetric function appearing in the shuffle theorem under any line, thus\u0000proving a triangular version of the $(q,t)$-Schr\"oder theorem.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141777254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Unicity Theorem and the center of the ${rm SL}_3$-skein algebra 统一性定理与 ${rm SL}_3$ skein 代数的中心
arXiv - MATH - Quantum Algebra Pub Date : 2024-07-23 DOI: arxiv-2407.16812
Hyun Kyu Kim, Zhihao Wang
{"title":"The Unicity Theorem and the center of the ${rm SL}_3$-skein algebra","authors":"Hyun Kyu Kim, Zhihao Wang","doi":"arxiv-2407.16812","DOIUrl":"https://doi.org/arxiv-2407.16812","url":null,"abstract":"The ${rm SL}_3$-skein algebra $mathscr{S}_{bar{q}}(mathfrak{S})$ of a\u0000punctured oriented surface $mathfrak{S}$ is a quantum deformation of the\u0000coordinate algebra of the ${rm SL}_3$-character variety of $mathfrak{S}$.\u0000When $bar{q}$ is a root of unity, we prove the Unicity Theorem for\u0000representations of $mathscr{S}_{bar{q}}(mathfrak{S})$, in particular the\u0000existence and uniqueness of a generic irreducible representation. Furthermore,\u0000we show that the center of $mathscr{S}_{bar{q}}(frak{S})$ is generated by\u0000the peripheral skeins around punctures and the central elements contained in\u0000the image of the Frobenius homomorphism for $mathscr{S}_{bar{q}}(frak{S})$,\u0000a surface generalization of Frobenius homomorphisms of quantum groups related\u0000to ${rm SL}_3$. We compute the rank of $mathscr{S}_{bar{q}}(mathfrak{S})$\u0000over its center, hence the dimension of the generic irreducible representation.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"81 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141777253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Non-semisimple $mathfrak{sl}_2$ quantum invariants of fibred links 纤维链路的非半纯 $mathfrak{sl}_2$ 量子不变式
arXiv - MATH - Quantum Algebra Pub Date : 2024-07-22 DOI: arxiv-2407.15561
Daniel López Neumann, Roland van der Veen
{"title":"Non-semisimple $mathfrak{sl}_2$ quantum invariants of fibred links","authors":"Daniel López Neumann, Roland van der Veen","doi":"arxiv-2407.15561","DOIUrl":"https://doi.org/arxiv-2407.15561","url":null,"abstract":"The Akutsu-Deguchi-Ohtsuki (ADO) invariants are the most studied quantum link\u0000invariants coming from a non-semisimple tensor category. We show that, for\u0000fibered links in $S^3$, the degree of the ADO invariant is determined by the\u0000genus and the top coefficient is a root of unity. More precisely, we prove that\u0000the top coefficient is determined by the Hopf invariant of the plane field of\u0000$S^3$ associated to the fiber surface. Our proof is based on the genus bounds\u0000established in our previous work, together with a theorem of Giroux-Goodman\u0000stating that fiber surfaces in the three-sphere can be obtained from a disk by\u0000plumbing/deplumbing Hopf bands.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141777255","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Projective geometries, $Q$-polynomial structures, and quantum groups 投影几何、Q$-多项式结构和量子群
arXiv - MATH - Quantum Algebra Pub Date : 2024-07-20 DOI: arxiv-2407.14964
Paul Terwilliger
{"title":"Projective geometries, $Q$-polynomial structures, and quantum groups","authors":"Paul Terwilliger","doi":"arxiv-2407.14964","DOIUrl":"https://doi.org/arxiv-2407.14964","url":null,"abstract":"In 2023 we obtained a $Q$-polynomial structure for the projective geometry\u0000$L_N(q)$. In the present paper, we display a more general $Q$-polynomial\u0000structure for $L_N(q)$. Our new $Q$-polynomial structure is defined using a\u0000free parameter $varphi$ that takes any positive real value. For $varphi=1$ we\u0000recover the original $Q$-polynomial structure. We interpret the new\u0000$Q$-polynomial structure using the quantum group $U_{q^{1/2}}(mathfrak{sl}_2)$\u0000in the equitable presentation. We use the new $Q$-polynomial structure to\u0000obtain analogs of the four split decompositions that appear in the theory of\u0000$Q$-polynomial distance-regular graphs.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141777256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Construction of Quantum Stabilizer Codes from Classical Codes and Butson Hadamard Matrices 从经典代码和 Butson Hadamard 矩阵构建量子稳定器代码
arXiv - MATH - Quantum Algebra Pub Date : 2024-07-18 DOI: arxiv-2407.13527
Bulent Sarac, Damla Acar
{"title":"A Construction of Quantum Stabilizer Codes from Classical Codes and Butson Hadamard Matrices","authors":"Bulent Sarac, Damla Acar","doi":"arxiv-2407.13527","DOIUrl":"https://doi.org/arxiv-2407.13527","url":null,"abstract":"In this paper, we give a constructive proof to show that if there exist a\u0000classical linear code C is a subset of F_q^n of dimension k and a classical\u0000linear code D is a subset of F_q^k^m of dimension s, where q is a power of a\u0000prime number p, then there exists an [[nm, ks, d]]_q quantum stabilizer code\u0000with d determined by C and D by identifying the stabilizer group of the code.\u0000In the construction, we use a particular type of Butson Hadamard matrices\u0000equivalent to multiple Kronecker products of the Fourier matrix of order p. We\u0000also consider the same construction of a quantum code for a general normalized\u0000Butson Hadamard matrix and search for a condition for the quantum code to be a\u0000stabilizer code.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"62 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741620","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Set-theoretic solutions of the Yang-Baxter equation and regular *-semibraces 杨-巴克斯特方程的集合论解和正则*振型
arXiv - MATH - Quantum Algebra Pub Date : 2024-07-17 DOI: arxiv-2407.12533
Qianxue Liu, Shoufeng Wang
{"title":"Set-theoretic solutions of the Yang-Baxter equation and regular *-semibraces","authors":"Qianxue Liu, Shoufeng Wang","doi":"arxiv-2407.12533","DOIUrl":"https://doi.org/arxiv-2407.12533","url":null,"abstract":"As generalizations of inverse semibraces introduced by Catino, Mazzotta and\u0000Stefanelli, Miccoli has introduced regular $star$-semibraces under the name of\u0000involution semibraces and given a sufficient condition under which the\u0000associated map to a regular $star$-semibrace is a set-theoretic solution of\u0000the Yang-Baxter equation. From the viewpoint of universal algebra, regular\u0000$star$-semibraces are (2,2,1)-type algebras. In this paper we continue to\u0000study set-theoretic solutions of the Yang-Baxter equation and regular\u0000$star$-semibraces. We first consider several kinds of (2,2,1)-type algebras\u0000that induced by regular $star$-semigroups and give some equivalent\u0000characterizations of the statement that they form regular $star$-semibraces.\u0000Then we give sufficient and necessary conditions under which the associated\u0000maps to these (2,2,1)-type algebras are set-theoretic solutions of the\u0000Yang-Baxter equation. Finally, as analogues of weak braces defined by Catino,\u0000Mazzotta, Miccoli and Stefanelli, we introduce weak $star$-braces in the class\u0000of regular $star$-semibraces, describe their algebraic structures and prove\u0000that the associated maps to weak $star$-braces are always set-theoretic\u0000solutions of the Yang-Baxter equation. The result of the present paper shows\u0000that the class of completely regular, orthodox and locally inverse regular\u0000$star$-semigroups is a source of possibly new set-theoretic solutions of the\u0000Yang-Baxter equation. Our results establish the close connection between the\u0000Yang-Baxter equation and the classical structural theory of regular\u0000$star$-semigroups.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Quantum geometric Wigner construction for $D(G)$ and braided racks D(G)$和编织架的量子几何维格纳构造
arXiv - MATH - Quantum Algebra Pub Date : 2024-07-16 DOI: arxiv-2407.11835
Shahn Majid, Leo Sean McCormack
{"title":"Quantum geometric Wigner construction for $D(G)$ and braided racks","authors":"Shahn Majid, Leo Sean McCormack","doi":"arxiv-2407.11835","DOIUrl":"https://doi.org/arxiv-2407.11835","url":null,"abstract":"The quantum double $D(G)=Bbb C(G)rtimes Bbb C G$ of a finite group plays\u0000an important role in the Kitaev model for quantum computing, as well as in\u0000associated TQFT's, as a kind of Poincar'e group. We interpret the known\u0000construction of its irreps, which are quasiparticles for the model, in a\u0000geometric manner strictly analogous to the Wigner construction for the usual\u0000Poincar'e group of $Bbb R^{1,3}$. Irreps are labelled by pairs $(C, pi)$,\u0000where $C$ is a conjugacy class in the role of a mass-shell, and $pi$ is a\u0000representation of the isotropy group $C_G$ in the role of spin. The geometric\u0000picture entails $D^vee(G)to Bbb C(C_G)blacktriangleright!!!!< Bbb C G$\u0000as a quantum homogeneous bundle where the base is $G/C_G$, and $D^vee(G)to\u0000Bbb C(G)$ as another homogeneous bundle where the base is the group algebra\u0000$Bbb C G$ as noncommutative spacetime. Analysis of the latter leads to a\u0000duality whereby the differential calculus and solutions of the wave equation on\u0000$Bbb C G$ are governed by irreps and conjugacy classes of $G$ respectively,\u0000while the same picture on $Bbb C(G)$ is governed by the reversed data.\u0000Quasiparticles as irreps of $D(G)$ also turn out to classify irreducible\u0000bicovariant differential structures $Omega^1_{C, pi}$ on $D^vee(G)$ and\u0000these in turn correspond to braided-Lie algebras $mathcal{L}_{C, pi}$ in the\u0000braided category of $G$-crossed modules, which we call `braided racks' and\u0000study. We show under mild assumptions that $U(mathcal{L}_{C,pi})$ quotients\u0000to a braided Hopf algebra $B_{C,pi}$ related by transmutation to a\u0000coquasitriangular Hopf algebra $H_{C,pi}$.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141722078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Twisted edge Laplacians on finite graphs from a Kähler structure 从凯勒结构看有限图上的扭曲边拉普拉奇
arXiv - MATH - Quantum Algebra Pub Date : 2024-07-16 DOI: arxiv-2407.11400
Soumalya Joardar, Atibur Rahaman
{"title":"Twisted edge Laplacians on finite graphs from a Kähler structure","authors":"Soumalya Joardar, Atibur Rahaman","doi":"arxiv-2407.11400","DOIUrl":"https://doi.org/arxiv-2407.11400","url":null,"abstract":"In this paper we study a Kahler structure on finite points. In particular, we\u0000study the edge Laplacian of a graph twisted by the Kahler structure introduced\u0000in this paper. We also discuss a metric aspect from a twisted holomorphic\u0000Dolbeault-Dirac spectral triple and show that the points have a finite diameter\u0000with respect to Connes' distance.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718012","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the higher-rank Askey-Wilson algebras 关于高阶阿斯基-威尔逊代数
arXiv - MATH - Quantum Algebra Pub Date : 2024-07-15 DOI: arxiv-2407.10404
Wanxia Wang, Shilin Yang
{"title":"On the higher-rank Askey-Wilson algebras","authors":"Wanxia Wang, Shilin Yang","doi":"arxiv-2407.10404","DOIUrl":"https://doi.org/arxiv-2407.10404","url":null,"abstract":"In the paper, a new algebra ${mathcal A}(n)$, which is generated by an upper\u0000triangular generating matrix with triple relations, is introduced. It is shown\u0000that there exists an isomorphism between the algebra ${mathcal A}(n)$ and the\u0000higher Askey-Wilson algebra ${mathfrak{aw}}(n)$ introduced by Cramp'{e},\u0000Frappat et al. Furthermore, we establish a series of automorphisms of\u0000${mathcal A}(n),$ which satisfy braid group relations and coincide with those\u0000in ${mathfrak{aw}}(n).$","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718050","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Computing the Khovanov homology of 2 strand braid links via generators and relations 通过生成器和关系计算双股辫状链的科瓦诺夫同源性
arXiv - MATH - Quantum Algebra Pub Date : 2024-07-13 DOI: arxiv-2407.09785
Domenico Fiorenza, Omid Hurson
{"title":"Computing the Khovanov homology of 2 strand braid links via generators and relations","authors":"Domenico Fiorenza, Omid Hurson","doi":"arxiv-2407.09785","DOIUrl":"https://doi.org/arxiv-2407.09785","url":null,"abstract":"In \"Homfly polynomial via an invariant of colored plane graphs\", Murakami,\u0000Ohtsuki, and Yamada provide a state-sum description of the level $n$ Jones\u0000polynomial of an oriented link in terms of a suitable braided monoidal category\u0000whose morphisms are $mathbb{Q}[q,q^{-1}]$-linear combinations of oriented\u0000trivalent planar graphs, and give a corresponding description for the HOMFLY-PT\u0000polynomial. We extend this construction and express the Khovanov-Rozansky\u0000homology of an oriented link in terms of a combinatorially defined category\u0000whose morphisms are equivalence classes of formal complexes of (formal direct\u0000sums of shifted) oriented trivalent plane graphs. By working combinatorially,\u0000one avoids many of the computational difficulties involved in the matrix\u0000factorization computations of the original Khovanov-Rozansky formulation: one\u0000systematically uses combinatorial relations satisfied by these matrix\u0000factorizations to simplify the computation at a level that is easily handled.\u0000By using this technique, we are able to provide a computation of the level $n$\u0000Khovanov-Rozansky invariant of the 2-strand braid link with $k$ crossings, for\u0000arbitrary $n$ and $k$, confirming and extending previous results and\u0000conjectural predictions by Anokhina-Morozov, Beliakova-Putyra-Wehrli,\u0000Carqueville-Murfet, Dolotin-Morozov, Gukov-Iqbal-Kozcaz-Vafa,\u0000Nizami-Munir-Sohail-Usman, and Rasmussen.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718014","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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