Crystallizing the hypoplactic monoid

Crystallizing the hypoplactic monoid António Malheiro & Alan Cain CMA/FCT Universidade Nova de Lisboa July 4, 06 Acknowledgement: This work was supported by CMA within the project UID/MAT/0097/03 financed by Fundação para a Ciência e a Tecnologia

Hypoplactic: a kind of plactic. The plactic monoid has applications to... representation theory; algebraic combinatorics; algebraic geometry.

Hypoplactic: a kind of plactic. The plactic monoid has applications to... representation theory; algebraic combinatorics; algebraic geometry. Plactic monoid Young tableaux 3 4 3 Knuth relations (acb, cab) : a b < c; (bac, bca) : a < b c Crystals 3 3 33 Hypoplactic monoid Quasi-ribbon tableaux 3 3 4 Knuth + (cadb, acbd) : a b < c d; (bdac, dbca) : a < b c < d?

Young tableaux 3 4 3 6 5 Young tableau: Rows non-decreasing left to right. Columns increasing top to bottom. Longer columns to the left. 3 5 6 8 4 9 7 Standard Young tableau: contains each symbol in {,..., n} (for some n) exactly once.

Young tableaux 3 4 3 6 5 3 Schensted s algorithm inserts a symbol a N into a tableau T:. If appending a to the end of the top row gives a tableau, this is the result.. Otherwise, let b the leftmost symbol of the top row such that b > a. Replace b with a ( bumping b ). 3. Recursively insert b into the tableau formed by all rows below the topmost. Result is denoted T a.

Young tableaux 3 3 3 6 4 5 Schensted s algorithm inserts a symbol a N into a tableau T:. If appending a to the end of the top row gives a tableau, this is the result.. Otherwise, let b the leftmost symbol of the top row such that b > a. Replace b with a ( bumping b ). 3. Recursively insert b into the tableau formed by all rows below the topmost. Result is denoted T a.

Young tableaux 3 3 3 4 5 6 Schensted s algorithm inserts a symbol a N into a tableau T:. If appending a to the end of the top row gives a tableau, this is the result.. Otherwise, let b the leftmost symbol of the top row such that b > a. Replace b with a ( bumping b ). 3. Recursively insert b into the tableau formed by all rows below the topmost. Result is denoted T a.

Robinson Schensted correspondence Let A n = { < <... < n}. From a word w = w w n A n : compute a tableau P(w) by inserting w,..., w n. compute a standard tableau Q(w) (the recording tableau) by labelling in order the new squares. Computing the pair ( P(w), Q(w) ) from w = 33: ( ) ( ) (, ) 33 =, 33 = 3, 3 ( ) ( ) = 3, 3 = 3 3, 3 4 3 ( ) 3 5 3 5 =, =, 3 4. 3 3 4 3 6 From a pair (P, Q) of tableau of the same shape, with Q standard, we can reverse this process.

Robinson Schensted correspondence Theorem (Robinson 938, Schensted 96) The map w ( P(w), Q(w) ) is a bijection. By holding P and varying Q over all standard tableau of the same shape, we get all words w such that P(w) = P. For example: ( ( ( 3 3 3,,, 3 4 4 3 3 4 ) ) ) 3, 3, 3.

Column readings 3 3 4 5 6 Column reading of a tableau: From leftmost column to rightmost. In each column, from bottom to top. Example gives 56343. Result is a tableau word.

Column readings 3 3 4 5 6 4 7 9 5 8 3 6 Column reading of a tableau: From leftmost column to rightmost. In each column, from bottom to top. Example gives 56343. Result is a tableau word. Corresponds to taking recording tableau of this form, where columns are filled from top to bottom with consecutive numbers.

Plactic monoid Theorem (Knuth 970) The relation = plac defined by u = plac v P(u) = P(v) is a congruence on A n. The factor monoid plac n = A n/= plac is the Plactic monoid of rank n. plac n is presented by A n Rplac, where R plac = { (acb, cab) : a b < c } Tableaux form a cross-section. { (bac, bca) : a < b c }. Indexes representations of the special linear Lie algebra sl n+. Analogous plactic monoids corresponding to other Lie algebras sp n, so n+, so n, and the exceptional Lie algebra G. Each of these monoids has a notion of tableaux and an insertion algorithm.

Crystal graph for plac n ε 3 3 3 3 3 3 3 3 33 33 3 33 33 33 333 3 In purely combinatorial, monoid-theoretic terms: 3 3 3 3 33 A crystal graph is a labelled directed graph, with vertices being words in the free monoid A n. Isomorphisms between connected components correspond to the congruence = plac on A n. Isomorphism means weight-preserving labelled digraph isomorphism 33 3 3 3

Kashiwara operators The definition of the (partial) Kashiwara operators ẽ i and f i starts from the crystal basis for plac n : 3 n n 3... n n Define ẽ i, f i on A n by a i f i (a) and ẽ i (a) i a. Define on A n \ A n by { { u ẽ i (v) if ϕ i (u) < ɛ i (v) f ẽ i (uv) = ẽ i (u) v if ϕ i (u) ɛ i (v) ; i (u) v if ϕ i (u) > ɛ i (v) f i (uv) = u f i (v) if ϕ i (u) ɛ i (v) where ɛ i and ϕ i are auxiliary maps defined by ɛ i (w) = max { k N {0} : ẽ k i (w) is defined } ; ϕ i (w) = max { k N {0} : f k i (w) is defined }.

Computing Kashiwara operators To compute ẽ i (w) and f i (w): Replace each symbol a of w with + if a = i, if a = i +, ε if a / {i, i + }. Cancel subwords + (not subwords + ). Result is + φ i(w) ɛ i(w). ẽ i (w) is obtained by applying ẽ i to the symbol that was replaced by the leftmost remaining (if present). f i (w) is obtained by applying f i to the symbol that was replaced by the rightmost remaining + (if present). Let w = 3333 A 3. Computing ẽ (w) and f (w): 3 3 3 3

Building the crystal graph The operators ẽ i and f i are mutually inverse when defined: if ẽ i (w) is defined, then w = f i ẽ i (w); if f i (w) is defined, then w = ẽ i f i (w). The crystal graph for plac n is defined by: vertex set A n edges w i f i (w) whenever f i (w) is defined. The weight of a word w is the tuple describing the number of each symbol: wt(3333) = (4, 5, 4). Weights are ordered: ( (α,..., α l ) (β,..., β l ) ( k) α i β i ). i<k i<k The operators ẽ i and f i respecyively increase and decrease weights: wt(ẽ (3333)) = (4, 6, 3) wt( f (3333)) = (4, 4, 5).

Some isomorphic components of the graph for plac 3 3 3 3 3 3 3 3 3 3 3 33 3 3 33 3 3 33 3 33 33 3 33 33 3 33 33 3 333 33 333 33 333 33 333 333 333

Some isomorphic components of the graph for plac 3 3 3 3 3 3 3 3 3 3 3 3 33 3 3 33 3 3 33 3 33 33 3 33 33 3 33 33 3 333 33 333 33 333 33 333 333 333

Some isomorphic components of the graph for plac 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 33 3 3 33 3 3 3 3 3 3 33 33 3 33 33 3 3 3 3 3 3 333 33 333 33 3 3 3 333 333

Some isomorphic components of the graph for plac 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 33 3 3 33 3 3 3 3 3 3 33 33 3 33 33 3 3 3 3 3 3 333 33 333 33 3 3 3 333 333 }{{}}{{}}{{} Q = 3 4 4 3 3 4

An application of crystals Theorem (Cain, Gray, M. 04) Plactic monoids (and their analogies for sp n, so n+, so n, and the exceptional Lie algebra G ) admit presentations via finite complete rewriting systems.

Quasi-ribbon tableaux 3 4 4 6 8 Quasi-ribbon tableau: Top symbol of each column aligns with the bottom one on its left. Rows non-decreasing left to right. Columns increasing top to bottom.

Quasi-ribbon tableaux 3 4 4 6 8 Quasi-ribbon tableau: Top symbol of each column aligns with the bottom one on its left. Rows non-decreasing left to right. Columns increasing top to bottom. Column reading C(T): Leftmost column to rightmost. In each column, bottom to top. Example gives 43 64 8. Result is a quasi-ribbon word.

Insertion into quasi-ribbon tableaux 3 4 4 6 8 Krob Thibon algorithm inserts a symbol a N into a quasi-ribbon tableau T:. If a is greater than every symbol in T, add a to the right-hand end of T.. If a is less than every symbol in T, add a to the top of T. 3. Otherwise, let x the rightmost and bottommost symbol of T such that x a. Remove the part of T below and right of x, add a to the right of x, and replace the removed part below a. Result is denoted T a.

Insertion into quasi-ribbon tableaux 3 4 4 6 8 5 Krob Thibon algorithm inserts a symbol a N into a quasi-ribbon tableau T:. If a is greater than every symbol in T, add a to the right-hand end of T.. If a is less than every symbol in T, add a to the top of T. 3. Otherwise, let x the rightmost and bottommost symbol of T such that x a. Remove the part of T below and right of x, add a to the right of x, and replace the removed part below a. Result is denoted T a.

Insertion into quasi-ribbon tableaux 3 4 4 5 6 8 Krob Thibon algorithm inserts a symbol a N into a quasi-ribbon tableau T:. If a is greater than every symbol in T, add a to the right-hand end of T.. If a is less than every symbol in T, add a to the top of T. 3. Otherwise, let x the rightmost and bottommost symbol of T such that x a. Remove the part of T below and right of x, add a to the right of x, and replace the removed part below a. Result is denoted T a.

Quasi-ribbon tableaux From a word w w n A n : compute a quasi-ribbon tableau QRT(w) by inserting w,..., w n. compute a recording ribbon RR(w) by labelling in order the squares where symbols are inserted. Computing the pair ( QRT(w), RR(w) ) from w = 434: ( (, ) 434 = = = ( 4, 3 3 4 4, 3 ) ( ), 434 = 4, 34 ) 3 34 = 3, 4 4 4 5 4 = 3 Theorem The map w ( QRT(w), RR(w) ) is a bijection. 4 4, 3 6 4 5.

Hypoplactic monoid Define = hypo on A n by u = hypo v QRT(u) = QRT(v). Theorem (Krob Thibon 999, Novelli 000) The relation = hypo is a congruence on A n. The factor monoid hypo n = A n/= hypo is the hypoplactic monoid of rank n hypo n is presented by An Rplac R hypo, where R plac = { (acb, cab) : a b < c } { (bac, bca) : a < b c } ; R hypo = { (cadb, acbd) : a b < c d } { (bdac, dbca) : a < b c < d }. Quasi-ribbon tableaux form a cross-section.

Properties of crystals For the plactic monoid: Kashiwara operators ẽ i and f i raise and lower weights. ẽ i and f i preserve shapes of tableaux. Every connected component contains a unique highest-weight word. All tableaux of a given shape are in a single component. Isomorphisms of connected components correspond to = plac. Connected components are indexed by recording tableaux. Question Does the hypoplactic monoid admit a quasi-crystal structure with these features? (Mutatis mutandis)

Quasi-Kashiwara operators 3 n n 3... n n Define ë i, f i on A n by a i f i (a) and ë i (a) i a. For w A n \ A n, compute ë i (w) and f i (w): Replace each symbol a of w with + if a = i, if a = i +, ε if a / {i, i + }. If there is a subword +, then ë i (w) and f i (w) are undefined. ë i (w) is obtained by applying ë i to the symbol that was replaced by the leftmost (if present). f i (w) is obtained by applying f i to the symbol that was replaced by the rightmost + (if present). Let w = 33. Computing ë (w) and f (w): 3 3

Quasi-crystal graph for hypo 4 3 3 3 3 3 3 33 4 33 4 33 4 33 4 33 4 33 34 33 34 33 34 33 34 33 34 33 34 44 33 34 44 33 34 44 33 34 44 33 34 44 34 44 34 44 34 44 34 44 34 44 44 434 44 434 44 443 44 443 44 443 434 434 443 443 443 3434 4334 3443 4343 4433

Quasi-crystal graph for hypo 4 3 3 3 3 3 3 3 3 4 33 4 33 4 33 4 33 4 3 3 3 4 33 34 33 34 33 34 33 34 3 3 3 4 4 4 33 34 44 33 34 44 33 34 44 33 34 44 3 4 4 4 34 44 34 44 34 44 34 44 3 4 4 4 4 44 434 44 443 44 443 44 443 3 4 4 434 443 443 443 3 3 4 4 4334 3443 4343 4433

A hook-length formula The so-called hook-length formula gives the number of standard Young tableaux of a given shape, and thus of the size of a plactic class corresponding to an element of that shape. Shapes of quasi-ribbon tableau are determined by compositions α = (α,..., α m ). The weight of α is α = α + + α m. The length of α is l(α) = m. β is coarser than α (β α) if it can be formed from α by merging consecutive parts; e.g. () (3, 8) (3, 6, ) (3,, 5, ).

Size of quasi-crystal components Theorem (Cain, M. 06) The number of quasi-ribbon tableaux of shape α and symbols from A n is {( n+ α l(α) ) if l(α) n n l(α) 0 if l(α) > n.

Size of quasi-crystal components Theorem (Cain, M. 06) The number of quasi-ribbon tableaux of shape α and symbols from A n is {( n+ α l(α) ) if l(α) n n l(α) 0 if l(α) > n. In the previous example n = 4, α = 4 and l(α) =. It gives ( 6 ) = 5.

Crystals and quasi-crystals 3 3 3 3 4 3 3 4 3 3 4 33 4 3 3 4 33 4 3 3 4 43 33 34 33 4 3 4 43 33 34 33 4 4 43 33 34 44 333 34 4 43 33 34 44 333 34 43 433 34 44 333 334 44 43 433 34 44 333 334 44 433 44 434 334 344 433 44 434 334 344 4333 434 344 444 3433 434 344 444 4334 444 3434 444 4344 4434

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