Identities in plactic, hypoplactic, sylvester, Baxter, and related monoids

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1 Identities in plactic, hypoplactic, sylvester, Baxter, and related monoids Alan J. Cain Centro de Matemática e Aplicações Faculdade de Ciências e Tecnologia Universidade Nova de Lisboa a policromática CORES POSITIVO / CINZA gotipo for reproduzido a mais NEGATIVO de dem ser usadas múltiplas variantes, erá consultar nas páginas INVESTIGADOR FCT Joint { work with various subsets of António Malheiro (CMA), Georg Klein, Łukasz Kubat, Jan Okniński (Warsaw), Fábio Silva (Lisboa) } 9 (representado ao lado) é a onal oficial e poderá ser utilizada comunicações institucionais alquer outra cor deverá contactar abinete de comunicação e FCT a fim de saber qual a cor a eu contexto em concreto, visto sido definidas estratégias para a stas cores que não não estejam Funded by UID/MAT/0097/0 Funded by IF/06/0/CP6/CT000

2 Young tableaux & Schensted s algorithm Let A = { < < <...} and let A n = { < < <... < n} Rows weakly increasing left to right. Columns strictly increasing top to bottom. Longer columns to the left. Schensted s algorithm To insert a A into a tableau:. If adding 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 ).. Recursively insert b into the tableau formed by all lower rows. For a word u = u u u k A : Start with an empty tableau and insert u, then u,..., finally u k. Call the resulting tableau P plac (u). For example, P plac (644) is the tableau above.

3 Young tableaux & Schensted s algorithm Let A = { < < <...} and let A n = { < < <... < n} Rows weakly increasing left to right. Columns strictly increasing top to bottom. Longer columns to the left. Schensted s algorithm To insert a A into a tableau:. If adding 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 ).. Recursively insert b into the tableau formed by all lower rows. For a word u = u u u k A : Start with an empty tableau and insert u, then u,..., finally u k. Call the resulting tableau P plac (u). For example, P plac (644) is the tableau above.

4 Young tableaux & Schensted s algorithm Let A = { < < <...} and let A n = { < < <... < n} Rows weakly increasing left to right. Columns strictly increasing top to bottom. Longer columns to the left. Schensted s algorithm To insert a A into a tableau:. If adding 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 ).. Recursively insert b into the tableau formed by all lower rows. For a word u = u u u k A : Start with an empty tableau and insert u, then u,..., finally u k. Call the resulting tableau P plac (u). For example, P plac (644) is the tableau above.

5 Young tableaux & Schensted s algorithm Let A = { < < <...} and let A n = { < < <... < n} Rows weakly increasing left to right. Columns strictly increasing top to bottom. Longer columns to the left. Schensted s algorithm To insert a A into a tableau:. If adding 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 ).. Recursively insert b into the tableau formed by all lower rows. For a word u = u u u k A : Start with an empty tableau and insert u, then u,..., finally u k. Call the resulting tableau P plac (u). For example, P plac (644) is the tableau above.

6 Young tableaux & Schensted s algorithm Let A = { < < <...} and let A n = { < < <... < n} Rows weakly increasing left to right. Columns strictly increasing top to bottom. Longer columns to the left. Schensted s algorithm To insert a A into a tableau:. If adding 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 ).. Recursively insert b into the tableau formed by all lower rows. For a word u = u u u k A : Start with an empty tableau and insert u, then u,..., finally u k. Call the resulting tableau P plac (u). For example, P plac (644) is the tableau above.

7 Young tableaux & Schensted s algorithm Let A = { < < <...} and let A n = { < < <... < n} Rows weakly increasing left to right. Columns strictly increasing top to bottom. Longer columns to the left. Schensted s algorithm To insert a A into a tableau:. If adding 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 ).. Recursively insert b into the tableau formed by all lower rows. For a word u = u u u k A : Start with an empty tableau and insert u, then u,..., finally u k. Call the resulting tableau P plac (u). For example, P plac (644) is the tableau above.

8 The plactic monoid Schensted s algorithm computes a tableau P plac (u) for u A. Define u plac v P plac (u) = P plac (v). Theorem (Knuth 970) The relation plac is a congruence on A. plac = A / plac is the plactic monoid. plac n = A n/ plac is the plactic monoid of rank n. Clearly, plac plac... plac n plac n+... plac. and plac = n N plac n.

9 Identities An identity is a formal equality u = v, where u, v X. A monoid M satisfies u = v if substituting any element of M for each symbol in X gives an equality that holds in M. For example, Any commutative monoid satisfies xy = yx. Any nilpotent group of class satisfies xyzyx = yxzxy [Neumann & Taylor 96]. An identity is trivial if u and v are the same word; otherwise non-trivial. xy = xy is trivial. xy = yx and xyzyx = yxzxy are non-trivial. Questions Does plac satisfy a non-trivial identity? Does each plac n satisfy a non-trivial identity?

10 Chinese monoid The Chinese monoid is also defined by an insertion algorithm. The Chinese monoid is related to the plactic monoid by its growth type. plac is isomorphic to the Chinese monoid of rank. Proposition (Jaszuńska & Okniński) The Chinese monoid embeds into a direct product of copies of the bicyclic monoid and the infinite cyclic group. Proposition (Adian) The bicyclic monoid satisfies xyyxxyxyyx = xyyxyxxyyx ( Adian s identity ). Corollary The Chinese monoid satisfies Adian s identity. Corollary plac satisfies Adian s identity.

11 Identities for plac Adian s identity xyyxxyxyyx = xyyxyxxyyx Proposition (Kubat & Okniński) plac satisfies pqqpqp = pqpqqp, where p and q are the left and right sides of Adian s identity. plac does not satisfy Adian s identity. Use detailed calculations using normal forms. Proposition (Izhakian) plac satisfies xyxy x yxyxyxy x y = xyxy x yyxxyxy x y. Use a (complicated) representation in the monoid of upper-triangular tropical matrices. Proposition (C., Klein, Kubat, Malheiro, Okniński) plac satisfies pqppq = pqqpq, where p and q are left and right sides of Adian s identity. Use a (simpler) representation in the monoid of upper-triangular tropical matrices.

12 Identities for plac and plac n Theorem (C., Klein, Kubat, Malheiro, Okniński) plac does not satisfy any non-trivial identity. Proposition (CKKMO) plac n does not satisfy any non-trivial identity of length less than or equal to n. Theorem (Schensted 96) { Length of the longest weakly Number of columns in P plac (u) = increasing subsequence of u; { Length of the longest strictly Number of rows in P plac (u) = decreasing subsequence of u.

13 Identities for plac and plac n Theorem (C., Klein, Kubat, Malheiro, Okniński) plac does not satisfy any non-trivial identity. Proposition (CKKMO) plac n does not satisfy any non-trivial identity of length less than or equal to n. Theorem (Schensted 96) { Length of the longest weakly Number of columns in P plac (u) = increasing subsequence of u; { Length of the longest strictly Number of rows in P plac (u) = decreasing subsequence of u.

14 Identities for plac and plac n Theorem (C., Klein, Kubat, Malheiro, Okniński) plac does not satisfy any non-trivial identity. Proposition (CKKMO) plac n does not satisfy any non-trivial identity of length less than or equal to n. Theorem (Schensted 96) { Length of the longest weakly Number of columns in P plac (u) = increasing subsequence of u; { Length of the longest strictly Number of rows in P plac (u) = decreasing subsequence of u.

15 Proof that plac n satisfies no identity of length at most n Suppose plac n satisfies u(x, y) = v(x, y) of length n. Assume u = u u j xu j+ u n v = v v j yv j+ v n Let s = n A n, t = (n j)(n j + ) n A n (miss out n j + ). So the tableaux P plac ( u(s, t) ) and Pplac ( v(s, t) ) are equal. Longest decreasing subsequences: In u(s, t): u u u j x u j+ u n u n

16 Proof that plac n satisfies no identity of length at most n Suppose plac n satisfies u(x, y) = v(x, y) of length n. Assume u = u u j xu j+ u n v = v v j yv j+ v n Let s = n A n, t = (n j)(n j + ) n A n (miss out n j + ). So the tableaux P plac ( u(s, t) ) and Pplac ( v(s, t) ) are equal. Longest decreasing subsequences: n j + n n n j n j + In u(s, t): u u u j x u j+ u n u n

17 Proof that plac n satisfies no identity of length at most n Suppose plac n satisfies u(x, y) = v(x, y) of length n. Assume u = u u j xu j+ u n v = v v j yv j+ v n Let s = n A n, t = (n j)(n j + ) n A n (miss out n j + ). So the tableaux P plac ( u(s, t) ) and Pplac ( v(s, t) ) are equal. Longest decreasing subsequences: n j + n n n j n j + In u(s, t): u u u j x u j+ u n u n In v(s, t): v v v j y v j+ v n v n

18 Proof that plac n satisfies no identity of length at most n Suppose plac n satisfies u(x, y) = v(x, y) of length n. Assume u = u u j xu j+ u n v = v v j yv j+ v n Let s = n A n, t = (n j)(n j + ) n A n (miss out n j + ). So the tableaux P plac ( u(s, t) ) and Pplac ( v(s, t) ) are equal. Longest decreasing subsequences: n j + n n n j n j + In u(s, t): u u u j x u j+ u n u n In v(s, t): v v v j y v j+ v n v n n n n j n j + n j +

19 Proof that plac n satisfies no identity of length at most n Suppose plac n satisfies u(x, y) = v(x, y) of length n. Assume u = u u j xu j+ u n v = v v j yv j+ v n Let s = n A n, t = (n j)(n j + ) n A n (miss out n j + ). So the tableaux P plac ( u(s, t) ) and Pplac ( v(s, t) ) are equal. Longest decreasing subsequences: n j + n n n j n j + In u(s, t): u u u j x u j+ u n u n In v(s, t): v v v j y v j+ v n v n n n n j n j + n j +

20 Plactic-like monoids A family of monoids whose elements can be viewed as combinatorial objects: Plactic monoid Hypoplactic monoid Sylvester monoid Young tableaux Quasi-ribbon tableaux Binary search trees 4 4 Taiga monoid Stalactite monoid Baxter monoid BSTs with multiplicities Stalactite tableaux Pairs of twin binary search trees 4 4 7, 4 4

21 Binary search trees and leaf insertion Binary search tree (BST): 4 To insert a into a BST T: Add a as a leaf node in the unique position that yields a BST. 6 4 For a word u = u k u k u A. Start with an empty BST and insert u, then u,..., finally u k. Call the resulting BST P sylv (u). For example, P sylv (464) is the tree above.

22 Binary search trees and leaf insertion Binary search tree (BST): 4 To insert a into a BST T: Add a as a leaf node in the unique position that yields a BST. 6 4 For a word u = u k u k u A. Start with an empty BST and insert u, then u,..., finally u k. Call the resulting BST P sylv (u). For example, P sylv (464) is the tree above.

23 Binary search trees and leaf insertion Binary search tree (BST): 4 To insert a into a BST T: Add a as a leaf node in the unique position that yields a BST. 6 4 For a word u = u k u k u A. Start with an empty BST and insert u, then u,..., finally u k. Call the resulting BST P sylv (u). For example, P sylv (464) is the tree above.

24 Binary search trees and leaf insertion Binary search tree (BST): 4 To insert a into a BST T: Add a as a leaf node in the unique position that yields a BST. 6 4 For a word u = u k u k u A. Start with an empty BST and insert u, then u,..., finally u k. Call the resulting BST P sylv (u). For example, P sylv (464) is the tree above.

25 The sylvester monoid The insertion algorithm computes a tree P sylv (u) for u A. Define u sylv v P sylv (u) = P sylv (v). Theorem (Hivert et al. 00) The relation sylv is a congruence on A. sylv = A / sylv is the sylvester monoid. sylv n = A n/ sylv is the sylvester monoid of rank n. Theorem (C., Malheiro) sylv satisfies the identity xyxy = yxxy. This is the unique shortest non-trivial identity satisfied by sylv. unique up to renaming variables and swapping the two sides.

26 The sylvester monoid The insertion algorithm computes a tree P sylv (u) for u A. Define u sylv v P sylv (u) = P sylv (v). Theorem (Hivert et al. 00) The relation sylv is a congruence on A. sylv = A / sylv is the sylvester monoid. sylv n = A n/ sylv is the sylvester monoid of rank n. Theorem (C., Malheiro) sylv satisfies the identity xyxy = yxxy. This is the unique shortest non-trivial identity satisfied by sylv. unique up to renaming variables and swapping the two sides.

27 Idea of proof of identity for sylv Want to show that P sylv (stst) = P sylv (tsst) for all s, t A. Suffices to prove that P sylv (pr) = P sylv (qr) for all p, q, r A, where p, q, r contain the same number of each symbol.

28 Idea of proof of identity for sylv Want to show that P sylv (stst) = P sylv (tsst) for all s, t A. Suffices to prove that P sylv (pr) = P sylv (qr) for all p, q, r A, where p, q, r contain the same number of each symbol. P sylv (67 76) = P sylv (67 76) =

29 Idea of proof of identity for sylv Want to show that P sylv (stst) = P sylv (tsst) for all s, t A. Suffices to prove that P sylv (pr) = P sylv (qr) for all p, q, r A, where p, q, r contain the same number of each symbol. P sylv (67 76) = P sylv (67 76) =

30 Idea of proof of identity for sylv Want to show that P sylv (stst) = P sylv (tsst) for all s, t A. Suffices to prove that P sylv (pr) = P sylv (qr) for all p, q, r A, where p, q, r contain the same number of each symbol. P sylv (67 76) = P sylv (67 76) =

31 Idea of proof of identity for sylv Want to show that P sylv (stst) = P sylv (tsst) for all s, t A. Suffices to prove that P sylv (pr) = P sylv (qr) for all p, q, r A, where p, q, r contain the same number of each symbol. P sylv (67 76) = P sylv (67 76) =

32 Idea of proof of identity for sylv Want to show that P sylv (stst) = P sylv (tsst) for all s, t A. Suffices to prove that P sylv (pr) = P sylv (qr) for all p, q, r A, where p, q, r contain the same number of each symbol. P sylv (67 76) = P sylv (67 76) =

33 Idea of proof of identity for sylv Want to show that P sylv (stst) = P sylv (tsst) for all s, t A. Suffices to prove that P sylv (pr) = P sylv (qr) for all p, q, r A, where p, q, r contain the same number of each symbol. P sylv (67 76) = P sylv (67 76) =

34 Idea of proof of identity for sylv Want to show that P sylv (stst) = P sylv (tsst) for all s, t A. Suffices to prove that P sylv (pr) = P sylv (qr) for all p, q, r A, where p, q, r contain the same number of each symbol. P sylv (67 76) = 6 P sylv (67 76) = 6

35 Idea of proof of identity for sylv Want to show that P sylv (stst) = P sylv (tsst) for all s, t A. Suffices to prove that P sylv (pr) = P sylv (qr) for all p, q, r A, where p, q, r contain the same number of each symbol. P sylv (67 76) = 6 7 P sylv (67 76) = 6 7

36 Idea of proof of identity for sylv Want to show that P sylv (stst) = P sylv (tsst) for all s, t A. Suffices to prove that P sylv (pr) = P sylv (qr) for all p, q, r A, where p, q, r contain the same number of each symbol. P sylv (67 76) = 6 7 P sylv (67 76) = 6 7

37 Idea of proof of identity for sylv Want to show that P sylv (stst) = P sylv (tsst) for all s, t A. Suffices to prove that P sylv (pr) = P sylv (qr) for all p, q, r A, where p, q, r contain the same number of each symbol. P sylv (67 76) = 6 7 P sylv (67 76) = 6 7

38 Idea of proof of identity for sylv Want to show that P sylv (stst) = P sylv (tsst) for all s, t A. Suffices to prove that P sylv (pr) = P sylv (qr) for all p, q, r A, where p, q, r contain the same number of each symbol. P sylv (67 76) = 6 7 P sylv (67 76) = 6 7

39 Idea of proof of identity for sylv Want to show that P sylv (stst) = P sylv (tsst) for all s, t A. Suffices to prove that P sylv (pr) = P sylv (qr) for all p, q, r A, where p, q, r contain the same number of each symbol. P sylv (67 76) = P sylv (67 76) = 6 7

40 Idea of proof of identity for sylv Want to show that P sylv (stst) = P sylv (tsst) for all s, t A. Suffices to prove that P sylv (pr) = P sylv (qr) for all p, q, r A, where p, q, r contain the same number of each symbol. P sylv (67 76) = P sylv (67 76) = 6 7

41 Idea of proof of identity for sylv Want to show that P sylv (stst) = P sylv (tsst) for all s, t A. Suffices to prove that P sylv (pr) = P sylv (qr) for all p, q, r A, where p, q, r contain the same number of each symbol. P sylv (67 76) = P sylv (67 76) = 6 7

42 Idea of proof of identity for sylv Want to show that P sylv (stst) = P sylv (tsst) for all s, t A. Suffices to prove that P sylv (pr) = P sylv (qr) for all p, q, r A, where p, q, r contain the same number of each symbol. P sylv (67 76) = P sylv (67 76) = 6 7

43 Idea of proof of identity for sylv Want to show that P sylv (stst) = P sylv (tsst) for all s, t A. Suffices to prove that P sylv (pr) = P sylv (qr) for all p, q, r A, where p, q, r contain the same number of each symbol. P sylv (67 76) = P sylv (67 76) = 6 7

44 Idea of proof of identity for sylv Want to show that P sylv (stst) = P sylv (tsst) for all s, t A. Suffices to prove that P sylv (pr) = P sylv (qr) for all p, q, r A, where p, q, r contain the same number of each symbol. P sylv (67 76) = P sylv (67 76) = 6 7 7

45 Idea of proof of identity for sylv Want to show that P sylv (stst) = P sylv (tsst) for all s, t A. Suffices to prove that P sylv (pr) = P sylv (qr) for all p, q, r A, where p, q, r contain the same number of each symbol. P sylv (67 76) = P sylv (67 76) =

46 Idea of proof of identity for sylv Want to show that P sylv (stst) = P sylv (tsst) for all s, t A. Suffices to prove that P sylv (pr) = P sylv (qr) for all p, q, r A, where p, q, r contain the same number of each symbol. P sylv (67 76) = P sylv (67 76) =

47 Quasi-ribbon tableaux & insertion Quasi-ribbon tableau (QRT): For a word u = u u u n A. To insert a symbol a into a quasi-ribbon tableau T: Break the tableau two parts: T is up to and including the bottom-right-most symbol r such that r a; the remainder is T >. Add a to the right of r. Attach T > to the bottom of a. Start with an empty QRT and insert u, then u,..., finally u n. Call the resulting QRT P hypo (u). For example, P hypo (447)is the QRT above. Lemma If i < j are symbols in u and there is no k in u with i < k < j, then: i and j are on the different rows of P hypo (u) In u, some symbol i is to the right of some symbol j

48 Quasi-ribbon tableaux & insertion Quasi-ribbon tableau (QRT): For a word u = u u u n A. To insert a symbol a into a quasi-ribbon tableau T: Break the tableau two parts: T is up to and including the bottom-right-most symbol r such that r a; the remainder is T >. Add a to the right of r. Attach T > to the bottom of a. Start with an empty QRT and insert u, then u,..., finally u n. Call the resulting QRT P hypo (u). For example, P hypo (447)is the QRT above. Lemma If i < j are symbols in u and there is no k in u with i < k < j, then: i and j are on the different rows of P hypo (u) In u, some symbol i is to the right of some symbol j

49 Quasi-ribbon tableaux & insertion Quasi-ribbon tableau (QRT): For a word u = u u u n A. To insert a symbol a into a quasi-ribbon tableau T: Break the tableau two parts: T is up to and including the bottom-right-most symbol r such that r a; the remainder is T >. Add a to the right of r. Attach T > to the bottom of a. Start with an empty QRT and insert u, then u,..., finally u n. Call the resulting QRT P hypo (u). For example, P hypo (447)is the QRT above. Lemma If i < j are symbols in u and there is no k in u with i < k < j, then: i and j are on the different rows of P hypo (u) In u, some symbol i is to the right of some symbol j

50 Quasi-ribbon tableaux & insertion Quasi-ribbon tableau (QRT): For a word u = u u u n A. To insert a symbol a into a quasi-ribbon tableau T: Break the tableau two parts: T is up to and including the bottom-right-most symbol r such that r a; the remainder is T >. Add a to the right of r. Attach T > to the bottom of a. Start with an empty QRT and insert u, then u,..., finally u n. Call the resulting QRT P hypo (u). For example, P hypo (447)is the QRT above. Lemma If i < j are symbols in u and there is no k in u with i < k < j, then: i and j are on the different rows of P hypo (u) In u, some symbol i is to the right of some symbol j

51 Quasi-ribbon tableaux & insertion Quasi-ribbon tableau (QRT): For a word u = u u u n A. To insert a symbol a into a quasi-ribbon tableau T: Break the tableau two parts: T is up to and including the bottom-right-most symbol r such that r a; the remainder is T >. Add a to the right of r. Attach T > to the bottom of a. Start with an empty QRT and insert u, then u,..., finally u n. Call the resulting QRT P hypo (u). For example, P hypo (447)is the QRT above. Lemma If i < j are symbols in u and there is no k in u with i < k < j, then: i and j are on the different rows of P hypo (u) In u, some symbol i is to the right of some symbol j

52 Quasi-ribbon tableaux & insertion Quasi-ribbon tableau (QRT): For a word u = u u u n A. To insert a symbol a into a quasi-ribbon tableau T: Break the tableau two parts: T is up to and including the bottom-right-most symbol r such that r a; the remainder is T >. Add a to the right of r. Attach T > to the bottom of a. Start with an empty QRT and insert u, then u,..., finally u n. Call the resulting QRT P hypo (u). For example, P hypo (447)is the QRT above. Lemma If i < j are symbols in u and there is no k in u with i < k < j, then: i and j are on the different rows of P hypo (u) In u, some symbol i is to the right of some symbol j

53 Quasi-ribbon tableaux & insertion Quasi-ribbon tableau (QRT): For a word u = u u u n A. To insert a symbol a into a quasi-ribbon tableau T: Break the tableau two parts: T is up to and including the bottom-right-most symbol r such that r a; the remainder is T >. Add a to the right of r. Attach T > to the bottom of a. Start with an empty QRT and insert u, then u,..., finally u n. Call the resulting QRT P hypo (u). For example, P hypo (447)is the QRT above. Lemma If i < j are symbols in u and there is no k in u with i < k < j, then: i and j are on the different rows of P hypo (u) In u, some symbol i is to the right of some symbol j

54 Quasi-ribbon tableaux & insertion Quasi-ribbon tableau (QRT): For a word u = u u u n A. To insert a symbol a into a quasi-ribbon tableau T: Break the tableau two parts: T is up to and including the bottom-right-most symbol r such that r a; the remainder is T >. Add a to the right of r. Attach T > to the bottom of a. Start with an empty QRT and insert u, then u,..., finally u n. Call the resulting QRT P hypo (u). For example, P hypo (447)is the QRT above. Lemma If i < j are symbols in u and there is no k in u with i < k < j, then: i and j are on the different rows of P hypo (u) In u, some symbol i is to the right of some symbol j

55 Quasi-ribbon tableaux & insertion Quasi-ribbon tableau (QRT): For a word u = u u u n A. To insert a symbol a into a quasi-ribbon tableau T: Break the tableau two parts: T is up to and including the bottom-right-most symbol r such that r a; the remainder is T >. Add a to the right of r. Attach T > to the bottom of a. Start with an empty QRT and insert u, then u,..., finally u n. Call the resulting QRT P hypo (u). For example, P hypo (447)is the QRT above. Lemma If i < j are symbols in u and there is no k in u with i < k < j, then: i and j are on the different rows of P hypo (u) In u, some symbol i is to the right of some symbol j

56 The hypoplactic monoid Define u hypo v P hypo (u) = P hypo (v). Theorem (Novelli 000) The relation hypo is a congruence on A. hypo = A / hypo is the hypoplactic monoid. hypo n = A n/ hypo is the hypoplactic monoid of rank n. Theorem (C., Malheiro) hypo satisfies the identities xyxy = xyyx = yxxy = yxyx; xxyx = xyxx. These are the unique shortest non-trivial identities satisfied by hypo. A QRT is determined by the number of each symbol it contains and which symbols are on the same rows. These are the length-4 identities where the two sides preserve these properties.

57 Summary table Monoid Symbol Identity In rank n Plactic plac None? Hypoplactic hypo xyxy = yxyx Y Sylvester sylv xyxy = yxxy Y Baxter baxt yxxyxy = yxyxxy Y Stalactic stal xyx = yxx Y Taiga taig xyx = yxx Y Left patience sorting lps None N Right patience sorting rps None Y Question Does plac n satisfy a non-trivial identity for n 4? Conjectured hierarchy of identities for plac n, length n. Lots of random examples checked in plac 4 using Sage.