Crystal monoids. Robert Gray (joint work with A. J. Cain and A. Malheiro) AAA90: 90th Workshop on General Algebra Novi Sad, 5th June 2015

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1 Crystal monods Robert Gray (jont work wth A. J. Can and A. Malhero) AAA90: 90th Workshop on General Algebra Nov Sad, 5th June 05

2 Plactc monod va Knuth relatons Defnton Let A n be the fnte ordered alphabet { < <... < n}. Let R be the set of defnng relatons: zxy = xzy and yzx = yxz x < y < z, xyx = xxy and xyy = yxy x < y. The Plactc monod Pl(A n ) s the monod defned by the presentaton A n R. That s, Pl(A n ) = A n/ where s the smallest congruence on the free monod A n contanng R. We call the Plactc congruence. The relatons n ths presentaton are called the Knuth relatons.

3 The Plactc monod Has orgns n work of Schensted (96) and Knuth (970) concerned wth combnatoral problems on Young tableaux. Later studed n depth by Lascoux and Shützenberger (98). Due to close relatons to Young tableaux, has become a tool n several aspects of representaton theory and algebrac combnatorcs. T = w(t) = Fact: The set of word readngs of tableaux s a set of normal forms for the elements of the Plactc monod. So Pl(A n ) s the monod of tableaux: Elements The set of all tableaux over A n = { < < < n}. Products Computed usng Schensted nserton algorthm.

4 Crystals Fg 8.4 from Hong and Kang s book An ntroducton to quantum groups and crystal bases.

5 Crystal graphs (followng Kashwara and Nakashma (994)) Idea: Defne a drected labelled dgraph Γ An wth the propertes: Vertex set = A n Each drected edge s labelled by a symbol from the label set I = {,,..., n }. For each vertex u A n every I there s at most one drected edge labelled by leavng u, and there s at most one drected edge labelled by enterng u, u v, w u If u v then u = v, so words n the same component have the same length as each other. In partcular, connected components are all fnte.

6 Buldng the crystal graph Γ An A n = { < <... < n} We begn by specfyng structure on the words of length one Ths s known as a Crystal bass. n n... n n Kashwara operators on letters For each {,..., n } we defne partal maps ẽ and f on the letters A n called the Kashwara crystal graph operators. For each edge we defne f (a) = b and ẽ (b) = a. a b,

7 Kashwara operators on words Let u A n and I. Are ẽ (u) or f (u) defned? If so what words do we obtan? Example wth A 3 = { < < 3} 3 a f (a) ẽ, (b) Let u = and let = I = {, }. b

8 Kashwara operators on words Let u A n and I. Are ẽ (u) or f (u) defned? If so what words do we obtan? Example wth A 3 = { < < 3} 3 a f (a) ẽ, (b) Let u = and let = I = {, }. b

9 Kashwara operators on words Let u A n and I. Are ẽ (u) or f (u) defned? If so what words do we obtan? Example wth A 3 = { < < 3} 3 a f (a) ẽ, (b) Let u = and let = I = {, }. b

10 Kashwara operators on words Let u A n and I. Are ẽ (u) or f (u) defned? If so what words do we obtan? Example wth A 3 = { < < 3} 3 a f (a) ẽ, (b) Let u = and let = I = {, }. b

11 Kashwara operators on words Let u A n and I. Are ẽ (u) or f (u) defned? If so what words do we obtan? Example wth A 3 = { < < 3} 3 a f (a) ẽ, (b) Let u = and let = I = {, }. b

12 Kashwara operators on words Let u A n and I. Are ẽ (u) or f (u) defned? If so what words do we obtan? Example wth A 3 = { < < 3} 3 a f (a) ẽ, (b) Let u = and let = I = {, }. b = f (u)

13 Kashwara operators on words Let u A n and I. Are ẽ (u) or f (u) defned? If so what words do we obtan? Example wth A 3 = { < < 3} 3 a f (a) ẽ, (b) Let u = and let = I = {, }. b = f (u) = ẽ (u)

14 The crystal graph Γ An Defnton The crystal graph Γ An s the drected labelled graph wth: Vertex set: A n Drected labelled edges: for u A n u f (u), ẽ (u) u Note: When defned ẽ ( f (u)) = u and f (ẽ (u)) = u.

15 Crystal graph components for A 3 = { < < 3} Word length one 3

16 Crystal graph components for A 3 = { < < 3} Word length one 3 Word length two

17 Crystal graph components for A 3 = { < < 3} Word length three

18 Plactc monod va crystals Defnton: Two connected components B(w) and B(w ) of Γ An are somorphc f there s a label-preservng dgraph somorphsm f : B(w) B(w ). Fact: In Γ An f B(w) = B(w ) then there s a unque somorphsm f : B(w) B(w ).

19 Plactc monod va crystals Defnton: Two connected components B(w) and B(w ) of Γ An are somorphc f there s a label-preservng dgraph somorphsm f : B(w) B(w ). Fact: In Γ An f B(w) = B(w ) then there s a unque somorphsm f : B(w) B(w ). Theorem (Kashwara and Nakashma (994)) Let Γ An be the crystal graph wth crystal bass Defne a relaton on A n by n n... n n u w an somorphsm f : B(u) B(w) wth f (u) = w. Then s the Plactc congruence and Pl(A n ) = A n/ s the Plactc monod.

20 Crystal graph components for A 3 = { < < 3} =

21 Where do crystals come from? J. Hong, S.-J. Kang, Introducton to Quantum Groups and Crystal Bases. Stud. Math., vol. 4, Amer. Math. Soc., Provdence, RI, 00. Take a nce Le algebra g e.g. a fnte-dmensonal semsmple Le algebra. Crystal bases are bases of U q (g)-modules satsfyng certan axoms. Uq(g) = quantum deformaton of unversal envelopng algebra U(g) (Drnfeld and Jmbo (985). Every crystal bass has the structure of a coloured dgraph (called a crystal graph). The structure of these coloured dgraphs has been explctly determned for certan semsmple Le algebras (specal lnear, specal orthogonal, symplectc, some exceptonal types). Crystal constructed usng Kashwara operators s a combnatoral tool for studyng representatons of U q (g).

22 Crystal bases and crystal monods Le algebra Crystal bass Monod type n n A n : sl... n n n+ n n n n B n : so... n 0 n... n+ n n n C n : sp... n n... n Pl(A n ) Pl(B n ) Pl(C n ) n n n n... n n n... D n : so n n n n Pl(D n ) G Pl(G )

23 Known results and our nterest Known results on crystals A n, B n, C n, D n, or G and ther crystal monods:. Crystal bases - combnatoral descrpton Kashwara and Nakashma (994).. Tableaux theory and Schensted-type nserton algorthms - Kashwara and Nakashma (994), Lecouvey (00, 003, 007). 3. Fnte presentatons for Pl(X) va Knuth-type relatons - Lecouvey (00, 003, 007). Theory we have been developng for these monods: 4. Fnte complete rewrtng systems Fnte presentaton wth ordered relatons u R v where each word converges w R w to unque normal form. 5. Automatc structures Regular language of normal forms such that a A a fnte automaton recognsng pars of normal forms that dffer by multplcaton by a.

24 Our results A. J. Can, R. D. Gray, A. Malhero Crystal bases, fnte complete rewrtng systems, and bautomatc structures for Plactc monods of types A n, B n, C n, D n, and G. arxv:math.gr/4.7040, 50 pages. Theorem For any X {A n, B n, C n, D n, G }, there s a fnte complete rewrtng system (Σ, T) that presents Pl(X). Theorem The monods Pl(A n ), Pl(B n ), Pl(C n ), Pl(D n ), and Pl(G ) are all bautomatc. Corollary The monods Pl(A n ), Pl(B n ), Pl(C n ), Pl(D n ), and Pl(G ) all have word problem solvable n quadratc tme.

25 Current and future work Further develop the theory of crystal monods n general We can obtan other examples (e.g. bcyclc monod s a crystal monod). They all have decdable word problem. Under what condtons do they admt fnte complete rewrtng systems / are automatc? What do our results say about the Plactc algebras of Lttelmann? P. Lttelmann, A Plactc Algebra for Semsmple Le Algebras. Advances n Mathematcs 4 (996), Investgate how our results mght be appled to gve new computatonal tools for workng wth crystals (e.g. usng rewrtng systems / fnte automata to compute wth crystals).