, The 2-polygraph ACol 2 (n) is a finite convergent presentation of the symplectic plactic monoid P n (C)
, One remarks that the length of h is smaller than h, then h h. Second case: let h = pc u c v q and h = pc w c w q, with p and q are in ACol 1 (n) * and c u , c v , c w and c w are in ACol 1 (n), where w and w are respectively the readings of the right and left columns of P(uv). h h. Since every application of a 2-cell of ACol 2 (n) yields a -preceding word, addition, we have b = z l
, Since z l x 1 , the tableau P(eb) consists of two columns, that we denote by s and s . Then there is a 2-cell ? e,b : c e c b ? c s c s . In the other hand, we have d = z q . . . z p+1 y p . . . y 1 , d = z p . . . z 1 , s = z l . . . z q+1 y q . . . y p+1 x p . . . x 1 and s = z q . . . z p+1 y p . . . y 1 . Hence a = s, d = s and d = b which yields the confluence the tableau P(uw) consists of two columns, then x 1 z l . In addition, z l is greater than each element of v then y j z l . Hence, in all cases, the tableau P(eb) consists of two columns. On the other hand, using Schensted's algorithm, we called ?-normal if it satisfies ?(w) = w. The normalisation determines a monoid via the defining relation w = ?(w). A normalisation (? 1 , ?) is quadratic if the ?-normality of a 1-cell in ? * 1 only depends on its factors of length two and if we can go from a 1-cell w to the 1-cell ?(w) in finitely many steps, each of which consists in applying ? to some factors of length two. The class of a quadratic normalisation is a pair (x, y) of positive integers which means that one obtains the normal form after at most x steps when starting from the left and y steps from the right. Let ? be the restriction of ? to the set of 1-cells of length two
, Using the notion of quadratic normalisation of monoids, our construction allows us to give a new proof of the termination of the 2-polygraph Col 2 (n) without considering the combinatorial properties of tableaux. Consider the map ? : Col 1 (n) * ? Col 1 (n) * sending a 1-cell in Col 1 (n) * to its unique corresponding tableau, Then (Col, vol.1
In this way, we reduce the coherent presentation Col 3 (n) of the monoid P n into the coherent presentation Col 3 (n) of P n , whose underlying 2-polygraph is Col 2 (n) and the 3-cells X u,v,t are those of Col 3 (n), but with (u) = 1. We reduce in 4.2.3 the coherent presentation Col 3 (n) into a coherent presentation PreCol 3 (n) of the plactic monoid P n , whose underlying 2-polygraph is PreCol 2 (n) defined in Chapter 2, Subsection 2.2.2. This reduction is given by a collapsible part defined by a set of 3-cells of Col 3 (n). In a final step, we apply three steps of homotopical reduction on the (3, 1)-polygraph Col 3 (n) ,
, Let ? be a (3, 1)-polygraph. A collapsible part of ? is a triple ? = (? 2 , ? 3 , ? 4 such that the following conditions are satisfied: i) every ? of every ? k is collapsible, that is, t k?1 well-founded order on the cells of ? such that, for every ? in every ? k
, The homotopical reduction of the (3, 1)-polygraph ? with respect to a collapsible part ? is the Tietze transformation, denoted by R ? , from the (3, 1)-category ? 3 to the (3, 1)-category freely generated by the (3, 1)-polygraph obtained from ? by removing the cells of ? and all the corresponding redundant cells
, ) into a coherent presentation of the monoid P n . The set of 2-cells of this coherent presentation is given by
On monomial algebras of finite global dimension, Trans. Amer. Math. Soc, vol.291, issue.1, p.16002, 1985. ,
, On the homology of associative algebras, Trans. Amer. Math. Soc, vol.296, issue.2, pp.641-659, 1986.
, Term rewriting and all that, 1998.
An insertion scheme for C n crystals, Progr. Math, vol.191, pp.1-48, 1999. ,
A Schensted-type correspondence for the symplectic group, J. Combin. Theory Ser. A, vol.43, issue.2, pp.320-328, 1986. ,
The diamond lemma for ring theory, Adv. in Math, vol.29, issue.2, pp.178-218, 1978. ,
New approaches to plactic monoid via Gröbner-Shirshov bases, J. Algebra, vol.423, pp.301-317, 2015. ,
Imbeddings into simple associative algebras, Algebra i Logika, vol.15, issue.2, p.245, 1976. ,
String-rewriting systems, Texts and Monographs in Computer Science, 1993. ,
, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, 1968.
The geometry of rewriting systems: a proof of the Anick-Groves-Squier theorem, Algorithms and classification in combinatorial group theory, Math. Sci. Res. Inst. Publ, vol.23, pp.137-163, 1989. ,
Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal (An Algorithm for Finding the Basis Elements in the Residue Class Ring Modulo a Zero Dimensional Polynomial Ideal), J. of Symbolic Computation, Special Issue on Logic, Mathematics, and Computer Science: Interactions, vol.41, pp.475-511, 1965. ,
, History and basic features of the critical-pair/completion procedure, Rewriting techniques and applications, vol.3, 1985.
Higher-dimensional word problem, Category theory and computer science, Lecture Notes in Comput. Sci, vol.530, pp.94-105, 1991. ,
, Higher-dimensional word problems with applications to equational logic, 4th Summer Conference on Category Theory and Computer Science, vol.115, pp.43-62, 1991.
Crystal monoids & crystal bases: rewriting systems and biautomatic structures for plactic monoids of types A n, 2015. ,
, Finite Gröbner-Shirshov bases for Plactic algebras and biautomatic structures for Plactic monoids, J. Algebra, vol.423, pp.37-53, 2015.
, Rewriting systems and biautomatic structures for Chinese, hypoplactic, and Sylvester monoids, Internat. J. Algebra Comput, vol.25, issue.1-2, 2015.
The Chinese monoid, Int. J. Algebra Comput, vol.11, issue.3, pp.301-334, 2001. ,
URL : https://hal.archives-ouvertes.fr/hal-00622609
Gröbner-Shirshov basis for the Chinese monoid, J. Algebra Appl, vol.7, issue.5, pp.623-628, 2008. ,
Representations of U q (gl(n, C)) at q = 0 and the Robinson-Shensted, Physics and mathematics of strings, World Sci. Publ, pp.185-211, 1990. ,
Über die Topologie des dreidimensionalen Raumes, Math. Ann, vol.69, issue.1, pp.137-168, 1910. ,
, Free resolutions via Gröbner bases, 2009.
, Gröbner bases for operads, Duke Math. J, vol.153, issue.2, pp.363-396, 2010.
Hopf algebras and the quantum Yang-Baxter equation, Dokl. Akad. Nauk SSSR, vol.283, issue.5, pp.1060-1064, 1985. ,
Plactic-growth-like monoids, Words, languages and combinatorics, II (Kyoto, 1992), World Sci. Publ, pp.124-142, 1994. ,
The syzygy problem, Ann. of Math, issue.2, pp.323-333, 1981. ,
, Über die charakteristischen Einheiten der symmetrischen Gruppe., Berl. Ber. 1903, pp.328-358, 1903.
, With applications to representation theory and geometry, vol.35, 1997.
, , 1991.
Coherent presentations of Artin monoids, Compos. Math, vol.151, issue.5, pp.957-998, 2015. ,
URL : https://hal.archives-ouvertes.fr/hal-00682233
One-skeleton galleries, the path model, and a generalization of Macdonald's formula for Hall-Littlewood polynomials, Int. Math. Res. Not. IMRN, issue.12, pp.2649-2707, 2012. ,
A proof of the Bender-Knuth conjecture, Pacific J. Math, vol.108, issue.1, pp.99-113, 1983. ,
, Quadratic normalisation in monoids, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01141226
Higher-dimensional categories with finite derivation type, Theory Appl. Categ, vol.22, issue.18, pp.420-478, 2009. ,
URL : https://hal.archives-ouvertes.fr/hal-00326974
, Coherence in monoidal track categories, Math. Structures Comput. Sci, vol.22, issue.6, pp.931-969, 2012.
, Higher-dimensional normalisation strategies for acyclicity, Adv. Math, vol.231, issue.3-4, pp.2294-2351, 2012.
, Identities among relations for higher-dimensional rewriting systems, Sémin. Congr, vol.26, pp.145-161, 2009.
, Polygraphs of finite derivation type, to appear in Math, Structures Comput. Sci, 2016.
A homotopical completion procedure with applications to coherence of monoids, 24th International Conference on Rewriting Techniques and Applications, vol.21, pp.223-238, 2013. ,
URL : https://hal.archives-ouvertes.fr/hal-00818253
Finite derivation type property on the Chinese monoid, Appl. Math. Sci. (Ruse), vol.4, issue.21-24, pp.1073-1080, 2010. ,
, Finite derivation type property on the Chinese monoid, Appl. Math. Sci. (Ruse), vol.4, issue.21-24, pp.1073-1080, 2010.
Finite convergent presentation of plactic monoid for type C, Internat. J. Algebra Comput, vol.25, issue.8, pp.1239-1263, 2015. ,
URL : https://hal.archives-ouvertes.fr/hal-01174477
, Column presentations of plactic monoids, 2016.
Knuth's coherent presentations of plactic monoids for type A, p.1359677, 2016. ,
The algebra of binary search trees, Theoret. Comput. Sci, vol.339, issue.1, pp.129-165, 2005. ,
URL : https://hal.archives-ouvertes.fr/hal-00622706
Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, vol.42, 2002. ,
A complete proof of correctness of the Knuth-Bendix completion algorithm, J. Comput. System Sci, vol.23, issue.1, pp.11-21, 1981. ,
URL : https://hal.archives-ouvertes.fr/inria-00076536
Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol.9, 1978. ,
, Representations of semisimple Lie algebras in the BGGcategory O, vol.94, 2008.
Semi Thue systems and generalized Church-Rosser properties, Bericht Nr. 92, Fachbereich Informatik, 1982. ,
, A note on a special one-rule semi-Thue system, Inform. Process. Lett, vol.21, issue.3, pp.135-140, 1985.
A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Math. Phys, vol.10, issue.1, pp.63-69, 1985. ,
Minimal resolutions via algebraic discrete Morse theory, Mem. Amer. Math. Soc, vol.197, issue.923, p.74, 2009. ,
, Quantum groups and their primitive ideals, vol.29, 1995.
A finite Thue system with decidable word problem and without equivalent finite canonical system, Theoret. Comput. Sci, vol.35, issue.2-3, pp.337-344, 1985. ,
Crystallizing the q-analogue of universal enveloping algebras, Proceedings of the International Congress of Mathematicians, vol.I, pp.791-797, 1991. ,
, Global crystal bases of quantum groups, Duke Math. J, vol.69, issue.2, pp.455-485, 1993.
, On crystal bases, Representations of groups, vol.16, pp.155-197, 1994.
, Similarity of crystal bases, Lie algebras and their representations, vol.194, pp.177-186, 1995.
Crystal graphs for representations of the q-analogue of classical Lie algebras, J. Algebra, vol.165, issue.2, pp.295-345, 1994. ,
Term rewriting systems, Handbook of Logic in Computer Science, vol.2, pp.1-117, 1992. ,
Permutations, matrices, and generalized Young tableaux, Pacific J. Math, vol.34, pp.709-727, 1970. ,
Simple word problems in universal algebras, Computational Problems in Abstract Algebra, Proc. Conf, pp.263-297, 1967. ,
Gröbner-Shirshov bases for plactic algebras, Algebra Colloq, vol.21, issue.4, pp.591-596, 2014. ,
Super RSK-algorithms and super plactic monoid, Int. J. Algebra Comput, vol.16, issue.2, pp.377-396, 2006. ,
A Quillen model structure for 2-categories, K-Theory, vol.26, pp.171-205, 2002. ,
, A Quillen model structure for bicategories, K-Theory, vol.33, issue.3, pp.185-197, 2004.
Algebra and geometry of rewriting, Appl. Categ. Structures, vol.15, issue.4, pp.415-437, 2007. ,
, Algebra, vol.100, issue.2, pp.462-557, 1986.
, Standard monomial theory, Proceedings of the Hyderabad Conference on Algebraic Groups, pp.279-322, 1989.
Crystal graphs and q-analogues of weight multiplicities for the root system A n, Lett. Math. Phys, vol.35, issue.4, pp.359-374, 1995. ,
, C. R. Acad. Sci. Paris Sér. A-B, vol.286, issue.7, pp.323-324, 1978.
, Noncommutative structures in algebra and geometric combinatorics, vol.109, pp.129-156, 1978.
, Schubert polynomials and the Littlewood-Richardson rule, Lett. Math. Phys, vol.10, issue.2-3, pp.111-124, 1985.
, Noncommutative Schubert polynomials, Funktsional. Anal. i Prilozhen, vol.23, issue.3, pp.63-64, 1989.
The Robinson-Schensted correspondence, crystal bases, and the quantum straightening at q = 0, Electron. J. Combin, vol.3, issue.2, 1996. ,
Schensted-type correspondence, plactic monoid, and jeu de taquin for type C n, J. Algebra, vol.247, issue.2, pp.295-331, 2002. ,
, Schensted-type correspondences and plactic monoids for types B n and D n, J. Algebraic Combin, vol.18, issue.2, pp.99-133, 2003.
, Kostka-Foulkes polynomials cyclage graphs and charge statistic for the root system C n, J. Algebraic Combin, vol.21, issue.2, pp.203-240, 2005.
, Branching rules, Kostka-Foulkes polynomials and q-multiplicities in tensor product for the root system B n , C n and D n, vol.9, pp.377-402, 2006.
, Combinatorics of crystal graphs and Kostka-Foulkes polynomials for the root systems B n , C n and D n, European J. Combin, vol.27, issue.4, pp.526-557, 2006.
Combinatorics of crystal graphs for the root systems of types A n, MSJ Mem, vol.17, pp.11-41, 2007. ,
A generalization of the Littlewood-Richardson rule, J. Algebra, vol.130, issue.2, pp.328-368, 1990. ,
Richardson rule for symmetrizable Kac-Moody algebras, Invent. Math, vol.116, issue.1-3, pp.329-346, 1994. ,
, Crystal graphs and Young tableaux, J. Algebra, vol.175, issue.1, pp.65-87, 1995.
, Paths and root operators in representation theory, Ann. of Math, issue.2, pp.499-525, 1995.
, A plactic algebra for semisimple Lie algebras, Adv. Math, vol.124, issue.2, pp.312-331, 1996.
Parastatistics algebra, Young tableaux and the super plactic monoid, Int. J. Geom. Methods Mod. Phys, vol.5, issue.8, pp.1295-1314, 2008. ,
Cohomology rings of the plactic monoid algebra via a Gröbner-Shirshov basis, J. Algebra Appl, vol.15, issue.5, p.30, 2016. ,
, Algebraic combinatorics on words, Encyclopedia of Mathematics and its Applications, vol.90, 2002.
Introduction to quantum groups, Progress in Mathematics, vol.110, 1993. ,
Rewriting systems and hochschild-mitchell homology, Electr. Notes Theor. Comput. Sci, vol.81, pp.59-72, 2003. ,
On the impossibility of certain algorithms in the theory of associative systems, Doklady Akad. Nauk SSSR (N.S.), vol.55, pp.583-586, 1947. ,
, On the impossibility of certain algorithms in the theory of associative systems, vol.58, pp.353-356, 1947.
, Resolutions by polygraphs, vol.11, pp.148-184, 2003.
An introduction to commutative and noncommutative Gröbner bases, Second International Colloquium on Words, Languages and Combinatorics, vol.134, pp.131-173, 1992. ,
On theories with a combinatorial definition of "equivalence, Ann. of Math, issue.2, pp.223-243, 1942. ,
On the hypoplactic monoid, Discrete Math, vol.217, issue.1-3, pp.315-336, 2000. ,
URL : https://hal.archives-ouvertes.fr/hal-00622670
Recursive unsolvability of a problem of Thue, J. Symbolic Logic, vol.12, pp.1-11, 1947. ,
A 2-categorical pasting theorem, J. Algebra, vol.129, issue.2, pp.439-445, 1990. ,
, An n-categorical pasting theorem, Category theory (Como, 1990), Lecture Notes in Math, vol.1488, pp.326-358, 1991.
On the Representations of the Symmetric Group, Amer. J. Math, vol.60, issue.3, pp.745-760, 1938. ,
Longest increasing and decreasing subsequences, Canad. J. Math, vol.13, pp.179-191, 1961. ,
, Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, vol.579, pp.59-113, 1976.
, Pour le monoïde plaxique, Math. Inform. Sci. Humaines, issue.140, pp.5-10, 1997.
The shifted plactic monoid, Math. Z, vol.266, issue.2, pp.363-392, 2010. ,
URL : https://hal.archives-ouvertes.fr/hal-01185400
A symplectic jeu de taquin bijection between the tableaux of King and of De Concini, Trans. Amer. Math. Soc, vol.351, issue.9, pp.3569-3607, 1999. ,
, Crystals for dummies, Notes, 2005.
Some algorithmic problems for Lie algebras, Sib. Mat. Zh, vol.3, pp.292-296, 1962. ,
Morse theory from an algebraic viewpoint, Trans. Amer. Math. Soc, vol.358, issue.1, pp.115-129, 2006. ,
Word problems and a homological finiteness condition for monoids, J. Pure Appl. Algebra, vol.49, issue.1-2, pp.201-217, 1987. ,
A finiteness condition for rewriting systems, Theoret. Comput. Sci, vol.131, issue.2, pp.271-294, 1994. ,
With a foreword by Gian-Carlo Rota, vol.1, 1999. ,
Limits indexed by category-valued 2-functors, J. Pure Appl. Algebra, vol.8, issue.2, pp.149-181, 1976. ,
, The algebra of oriented simplexes, J. Pure Appl. Algebra, vol.49, issue.3, pp.283-335, 1987.
Orthogonal tableaux and an insertion algorithm for SO(2n + 1), J. Combin. Theory Ser. A, vol.53, issue.2, pp.239-256, 1990. ,
Term rewriting systems, Cambridge Tracts in Theoretical Computer Science, vol.55, 2003. ,
On Schensted's construction and the multiplication of Schur functions, Adv. Math, vol.30, pp.8-32, 1978. ,
Probleme über Veränderungen von Zeichenreihen nach gegebenen Regeln, vol.10, pp.493-524, 1914. ,
Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten, Monatsh. Math. Phys, vol.19, issue.1, pp.1-118, 1908. ,
Combinatorial and asymptotic methods in algebra, Algebra, VI, Encyclopaedia Math. Sci, vol.57, pp.1-196, 1995. ,
The Littlewood-Richardson rule and related combinatorics, Interaction of combinatorics and representation theory, pp.95-145, 2001. ,
On Quantitative Substitutional Analysis, Proc. London Math. Soc, issue.1, p.255 ,