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# Crystals of Modified Nakajima Monomials

AUTHORS:

- Arthur Lubovsky: Initial version
- Ben Salisbury: Initial version

Let \(Y_{i,k}\), for \(i \in I\) and \(k \in \ZZ\), be a commuting set of variables, and let \(\boldsymbol{1}\) be a new variable which commutes with each \(Y_{i,k}\). (Here, \(I\) represents the index set of a Cartan datum.) One may endow the structure of a crystal on the set \(\widehat{\mathcal{M}}\) of monomials of the form

\[M = \prod_{(i,k) \in I\times \ZZ_{\ge0}} Y_{i,k}^{y_i(k)}\boldsymbol{1}.\]

Elements of \(\widehat{\mathcal{M}}\) are called modified Nakajima monomials. We will omit the \(\boldsymbol{1}\) from the end of a monomial if there exists at least one \(y_i(k) \neq 0\). The crystal structure on this set is defined by

\[\begin{split}\begin{aligned} \mathrm{wt}(M) &= \sum_{i\in I} \Bigl( \sum_{k\ge 0} y_i(k) \Bigr) \Lambda_i, \\ \varphi_i(M) &= \max\Bigl\{ \sum_{0\le j \le k} y_i(j) : k\ge 0 \Bigr\}, \\ \varepsilon_i(M) &= \varphi_i(M) - \langle h_i, \mathrm{wt}(M) \rangle, \\ k_f = k_f(M) &= \min\Bigl\{ k\ge 0 : \varphi_i(M) = \sum_{0\le j\le k} y_i(j) \Bigr\}, \\ k_e = k_e(M) &= \max\Bigl\{ k\ge 0 : \varphi_i(M) = \sum_{0\le j\le k} y_i(j) \Bigr\}, \end{aligned}\end{split}\]

where \(\{h_i : i \in I\}\) and \(\{\Lambda_i : i \in I \}\) are the simple coroots and fundamental weights, respectively. With a chosen set of integers \(C = (c_{ij})_{i\neq j}\) such that \(c_{ij}+c{ji} =1\), one defines

\[A_{i,k} = Y_{i,k} Y_{i,k+1} \prod_{j\neq i} Y_{j,k+c_{ji}}^{a_{ji}},\]

where \((a_{ij})\) is a Cartan matrix. Then

\[\begin{split}\begin{aligned} e_iM &= \begin{cases} 0 & \text{if } \varepsilon_i(M) = 0, \\ A_{i,k_e}M & \text{if } \varepsilon_i(M) > 0, \end{cases} \\ f_iM &= A_{i,k_f}^{-1} M. \end{aligned}\end{split}\]

It is shown in [KKS07] that the connected component of \(\widehat{\mathcal{M}}\) containing the element \(\boldsymbol{1}\), which we denote by \(\mathcal{M}(\infty)\), is crystal isomorphic to the crystal \(B(\infty)\).

Let \(\widetilde{\mathcal{M}}\) be \(\widehat{\mathcal{M}}\) as a set, and with crystal structure defined as on \(\widehat{\mathcal{M}}\) with the exception that

\[\begin{split}f_iM = \begin{cases} 0 & \text{if } \varphi_i(M) = 0, \\ A_{i,k_f}^{-1}M & \text{if } \varphi_i(M) > 0. \end{cases}\end{split}\]

Then Kashiwara [Kash03] showed that the connected component in \(\widetilde{\mathcal{M}}\) containing a monomial \(M\) such that \(e_iM = 0\), for all \(i \in I\), is crystal isomorphic to the irreducible highest weight crystal \(B(\mathrm{wt}(M))\).

WARNING:

> Monomial crystals depend on the choice of positive integers \(C = (c_{ij})_{i\neq j}\) satisfying the condition \(c_{ij}+c_{ji}=1\). We have chosen such integers uniformly such that \(c_{ij} = 1\) if \(i < j\) and \(c_{ij} = 0\) if \(i>j\).

REFERENCES:

[KKS07] | (1, 2) S.-J. Kang, J.-A. Kim, and D.-U. Shin. Modified Nakajima Monomials and the Crystal \(B(\infty)\). J. Algebra 308, pp. 524–535, 2007.

[Kash03] | (1, 2) M. Kashiwara. Realizations of Crystals. Combinatorial and geometric representation theory (Seoul, 2001), Contemp. Math. 325, Amer. Math. Soc., pp. 133–139, 2003.

class `sage.combinat.crystals.monomial_crystals.``CrystalOfNakajimaMonomials`(ct, La)

  Bases: `sage.combinat.crystals.monomial_crystals.InfinityCrystalOfNakajimaMonomials`

  Let \(\widetilde{\mathcal{M}}\) be \(\widehat{\mathcal{M}}\) as a set, and with crystal structure defined as on \(\widehat{\mathcal{M}}\) with the exception that

  \[\begin{split}f_iM = \begin{cases} 0 & \text{if } \varphi_i(M) = 0, \\ A_{i,k_f}^{-1}M & \text{if } \varphi_i(M) > 0. \end{cases}\end{split}\]

  Then Kashiwara [Kash03] showed that the connected component in \(\widetilde{\mathcal{M}}\) containing a monomial \(M\) such that \(e_iM = 0\), for all \(i \in I\), is crystal isomorphic to the irreducible highest weight crystal \(B(\mathrm{wt}(M))\).

  INPUT:

  - `ct` – a Cartan type
  - `La` – an element of the weight lattice

  EXAMPLES:

      sage: La = RootSystem("A2").weight_lattice().fundamental_weights()
      sage: M = crystals.NakajimaMonomials("A2",La[1]+La[2])
      sage: B = crystals.Tableaux("A2",shape=[2,1])
      sage: GM = M.digraph()
      sage: GB = B.digraph()
      sage: GM.is_isomorphic(GB,edge_labels=True)
      True

      sage: La = RootSystem("G2").weight_lattice().fundamental_weights()
      sage: M = crystals.NakajimaMonomials("G2",La[1]+La[2])
      sage: B = crystals.Tableaux("G2",shape=[2,1])
      sage: GM = M.digraph()
      sage: GB = B.digraph()
      sage: GM.is_isomorphic(GB,edge_labels=True)
      True

      sage: La = RootSystem("B2").weight_lattice().fundamental_weights()
      sage: M = crystals.NakajimaMonomials(['B',2],La[1]+La[2])
      sage: B = crystals.Tableaux("B2",shape=[3/2,1/2])
      sage: GM = M.digraph()
      sage: GB = B.digraph()
      sage: GM.is_isomorphic(GB,edge_labels=True)
      True

      sage: La = RootSystem(['A',3,1]).weight_lattice().fundamental_weights()
      sage: M = crystals.NakajimaMonomials(['A',3,1],La[0]+La[2])
      sage: B = crystals.GeneralizedYoungWalls(3,La[0]+La[2])
      sage: SM = M.subcrystal(max_depth=4)
      sage: SB = B.subcrystal(max_depth=4)
      sage: GM = M.digraph(subset=SM) # long time
      sage: GB = B.digraph(subset=SB) # long time
      sage: GM.is_isomorphic(GB,edge_labels=True) # long time
      True

      sage: La = RootSystem(['A',5,2]).weight_lattice().fundamental_weights()
      sage: LA = RootSystem(['A',5,2]).weight_space().fundamental_weights()
      sage: M = crystals.NakajimaMonomials(['A',5,2],3*La[0])
      sage: B = crystals.LSPaths(3*LA[0])
      sage: SM = M.subcrystal(max_depth=4)
      sage: SB = B.subcrystal(max_depth=4)
      sage: GM = M.digraph(subset=SM)
      sage: GB = B.digraph(subset=SB)
      sage: GM.is_isomorphic(GB,edge_labels=True)
      True

  `cardinality`()

    Return the cardinality of `self`.

    EXAMPLES:

        sage: La = RootSystem(['A',2]).weight_lattice().fundamental_weights()
        sage: M = crystals.NakajimaMonomials(['A',2],La[1])
        sage: M.cardinality()
        3

        sage: La = RootSystem(['D',4,2]).weight_lattice().fundamental_weights()
        sage: M = crystals.NakajimaMonomials(['D',4,2],La[1])
        sage: M.cardinality()
        +Infinity

class `sage.combinat.crystals.monomial_crystals.``CrystalOfNakajimaMonomialsElement`(parent, dict)

  Bases: `sage.combinat.crystals.monomial_crystals.NakajimaYMonomial`

  Element class for `CrystalOfNakajimaMonomials`.

  The \(f_i\) operators need to be modified from the version in `monomial_crystalsNakajimaYMonomial` in order to create irreducible highest weight realizations. This modified \(f_i\) is defined as

  \[\begin{split}f_iM = \begin{cases} 0 & \text{if } \varphi_i(M) = 0, \\ A_{i,k_f}^{-1}M & \text{if } \varphi_i(M) > 0. \end{cases}\end{split}\]

  EXAMPLES:

      sage: La = RootSystem(['A',5,2]).weight_lattice().fundamental_weights()
      sage: M = crystals.NakajimaMonomials(['A',5,2],3*La[0])
      sage: m = M.module_generators[0].f(0); m
      Y(0,0)^2 Y(0,1)^-1 Y(2,0)
      sage: TestSuite(m).run()

  `f`(i)

    Return the action of \(f_i\) on `self`.

    INPUT:

    - `i` – an element of the index set

    EXAMPLES:

        sage: La = RootSystem(['A',5,2]).weight_lattice().fundamental_weights()
        sage: M = crystals.NakajimaMonomials(['A',5,2],3*La[0])
        sage: m = M.module_generators[0]
        sage: [m.f(i) for i in M.index_set()]
        [Y(0,0)^2 Y(0,1)^-1 Y(2,0) , None, None, None]

class `sage.combinat.crystals.monomial_crystals.``InfinityCrystalOfNakajimaMonomials`(ct, category, elt_class)

  Bases: `sage.structure.parent.Parent`, `sage.structure.unique_representation.UniqueRepresentation`

  Let \(Y_{i,k}\), for \(i \in I\) and \(k \in \ZZ\), be a commuting set of variables, and let \(\boldsymbol{1}\) be a new variable which commutes with each \(Y_{i,k}\). (Here, \(I\) represents the index set of a Cartan datum.) One may endow the structure of a crystal on the set \(\widehat{\mathcal{M}}\) of monomials of the form

  \[M = \prod_{(i,k) \in I\times \ZZ_{\ge0}} Y_{i,k}^{y_i(k)}\boldsymbol{1}.\]

  Elements of \(\widehat{\mathcal{M}}\) are called modified Nakajima monomials. We will omit the \(\boldsymbol{1}\) from the end of a monomial if there exists at least one \(y_i(k) \neq 0\). The crystal structure on this set is defined by

  \[\begin{split}\begin{aligned} \mathrm{wt}(M) &= \sum_{i\in I} \Bigl( \sum_{k\ge 0} y_i(k) \Bigr) \Lambda_i, \\ \varphi_i(M) &= \max\Bigl\{ \sum_{0\le j \le k} y_i(j) : k\ge 0 \Bigr\}, \\ \varepsilon_i(M) &= \varphi_i(M) - \langle h_i, \mathrm{wt}(M) \rangle, \\ k_f = k_f(M) &= \min\Bigl\{ k\ge 0 : \varphi_i(M) = \sum_{0\le j\le k} y_i(j) \Bigr\}, \\ k_e = k_e(M) &= \max\Bigl\{ k\ge 0 : \varphi_i(M) = \sum_{0\le j\le k} y_i(j) \Bigr\}, \end{aligned}\end{split}\]

  where \(\{h_i : i \in I\}\) and \(\{\Lambda_i : i \in I \}\) are the simple coroots and fundamental weights, respectively. With a chosen set of integers \(C = (c_{ij})_{i\neq j}\) such that \(c_{ij}+c{ji} =1\), one defines

  \[A_{i,k} = Y_{i,k} Y_{i,k+1} \prod_{j\neq i} Y_{j,k+c_{ji}}^{a_{ji}},\]

  where \((a_{ij})\) is a Cartan matrix. Then

  \[\begin{split}\begin{aligned} e_iM &= \begin{cases} 0 & \text{if } \varepsilon_i(M) = 0, \\ A_{i,k_e}M & \text{if } \varepsilon_i(M) > 0, \end{cases} \\ f_iM &= A_{i,k_f}^{-1} M. \end{aligned}\end{split}\]

  It is shown in [KKS07] that the connected component of \(\widehat{\mathcal{M}}\) containing the element \(\boldsymbol{1}\), which we denote by \(\mathcal{M}(\infty)\), is crystal isomorphic to the crystal \(B(\infty)\).

  INPUT:

  - `cartan_type` – a Cartan type
  - `use_Y` – choice of monomials in terms of \(A\) or \(Y\)

  EXAMPLES:

      sage: B = crystals.infinity.Tableaux("C3")
      sage: S = B.subcrystal(max_depth=4)
      sage: G = B.digraph(subset=S) # long time
      sage: M = crystals.infinity.NakajimaMonomials("C3") # long time
      sage: T = M.subcrystal(max_depth=4) # long time
      sage: H = M.digraph(subset=T) # long time
      sage: G.is_isomorphic(H,edge_labels=True) # long time
      True

      sage: M = crystals.infinity.NakajimaMonomials(['A',2,1])
      sage: T = M.subcrystal(max_depth=3)
      sage: H = M.digraph(subset=T) # long time
      sage: Y = crystals.infinity.GeneralizedYoungWalls(2)
      sage: YS = Y.subcrystal(max_depth=3)
      sage: YG = Y.digraph(subset=YS) # long time
      sage: YG.is_isomorphic(H,edge_labels=True) # long time
      True

      sage: M = crystals.infinity.NakajimaMonomials("D4")
      sage: B = crystals.infinity.Tableaux("D4")
      sage: MS = M.subcrystal(max_depth=3)
      sage: BS = B.subcrystal(max_depth=3)
      sage: MG = M.digraph(subset=MS) # long time
      sage: BG = B.digraph(subset=BS) # long time
      sage: BG.is_isomorphic(MG,edge_labels=True) # long time
      True

  `cardinality`()

    Return the cardinality of `self`, which is always \(\infty\).

    EXAMPLES:

        sage: M = crystals.infinity.NakajimaMonomials(['A',5,2])
        sage: M.cardinality()
        +Infinity

class `sage.combinat.crystals.monomial_crystals.``NakajimaAMonomial`(parent, dict)

  Bases: `sage.combinat.crystals.monomial_crystals.NakajimaYMonomial`

  Monomials of the form \(A_{i_1,k_1}^{a_1}\cdots A_{i_t,k_t}^{a_t}\), where \(i_1,\dots,i_t\) are elements of the index set, \(k_1,\dots,k_t\) are nonnegative integers, and \(a_1,\dots,a_t\) are integers.

  EXAMPLES:

      sage: M = crystals.infinity.NakajimaMonomials("A3",use_Y=False)
      sage: mg = M.module_generators[0]
      sage: mg.f_string([1,2,3,2,1])
      A(1,0)^-1 A(1,1)^-1 A(2,0)^-2 A(3,0)^-1
      sage: mg.f_string([3,2,1])
      A(1,2)^-1 A(2,1)^-1 A(3,0)^-1

  `e`(i)

    Return the action of \(e_i\) on `self`.

    INPUT:

    - `i` – an element of the index set

    EXAMPLES:

        sage: M = crystals.infinity.NakajimaMonomials(['D',4,1],use_Y=False)
        sage: m = M.module_generators[0].f_string([4,2,3,0])
        sage: [m.e(i) for i in M.index_set()]
        [A(2,1)^-1 A(3,1)^-1 A(4,0)^-1 ,
         None,
         None,
         A(0,2)^-1 A(2,1)^-1 A(4,0)^-1 ,
         None]

  `epsilon`(i)

    Return the action of \(\varepsilon_i\) on `self`.

    INPUT:

    - `i` – an element of the index set

    EXAMPLES:

        sage: M = crystals.infinity.NakajimaMonomials(['C',4,1],use_Y=False)
        sage: m = M.module_generators[0].f_string([4,2,3])
        sage: [m.epsilon(i) for i in M.index_set()]
        [0, 0, 0, 1, 0]

  `f`(i)

    Return the action of \(f_i\) on `self`.

    INPUT:

    - `i` – an element of the index set

    EXAMPLES:

        sage: M = crystals.infinity.NakajimaMonomials("E8",use_Y=False)
        sage: m = M.module_generators[0].f_string([4,2,3,8])
        sage: m
        A(2,1)^-1 A(3,1)^-1 A(4,0)^-1 A(8,0)^-1
        sage: [m.f(i) for i in M.index_set()]
        [A(1,2)^-1 A(2,1)^-1 A(3,1)^-1 A(4,0)^-1 A(8,0)^-1 ,
         A(2,0)^-1 A(2,1)^-1 A(3,1)^-1 A(4,0)^-1 A(8,0)^-1 ,
         A(2,1)^-1 A(3,0)^-1 A(3,1)^-1 A(4,0)^-1 A(8,0)^-1 ,
         A(2,1)^-1 A(3,1)^-1 A(4,0)^-1 A(4,1)^-1 A(8,0)^-1 ,
         A(2,1)^-1 A(3,1)^-1 A(4,0)^-1 A(5,0)^-1 A(8,0)^-1 ,
         A(2,1)^-1 A(3,1)^-1 A(4,0)^-1 A(6,0)^-1 A(8,0)^-1 ,
         A(2,1)^-1 A(3,1)^-1 A(4,0)^-1 A(7,1)^-1 A(8,0)^-1 ,
         A(2,1)^-1 A(3,1)^-1 A(4,0)^-1 A(8,0)^-2 ]

  `phi`(i)

    Return the action of \(\varphi_i\) on `self`.

    INPUT:

    - `i` – an element of the index set

    EXAMPLES:

        sage: M = crystals.infinity.NakajimaMonomials(['C',4,1],use_Y=False)
        sage: m = M.module_generators[0].f_string([4,2,3])
        sage: [m.phi(i) for i in M.index_set()]
        [0, 1, -1, 2, -1]

  `to_Y_monomial`()

    Represent \(\prod_{(i,k)} A_{i,k}^{a_{i}(k)}\) in the form \(\prod_{(i,k)} Y_{i,k}^{y_i(k)}\) using the formula

    \[\begin{split}A_{i,k} = Y_{i,k} Y_{i,k+1} \prod_{\substack{j \in I \\ j\neq i}} Y_{i,k+c_{ji}}^{a_{ji}}.\end{split}\]

    EXAMPLES:

        sage: M = crystals.infinity.NakajimaMonomials(['A',2,1],use_Y=False)
        sage: m = M.module_generators[0].f_string([2,0,1,2,1])
        sage: m
        A(0,0)^-1 A(1,0)^-1 A(1,1)^-1 A(2,0)^-1 A(2,1)^-1
        sage: m.to_Y_monomial()
        Y(0,1) Y(0,2) Y(1,1)^-1 Y(2,2)^-1

  `weight`()

    Return the weight of `self` as an element of `self.parent().weight_lattice_realization()`.

    EXAMPLES:

        sage: M = crystals.infinity.NakajimaMonomials(['A',4,2],use_Y=False)
        sage: m = M.module_generators[0].f_string([1,2,0,1])
        sage: m.weight()
        2*Lambda[0] - Lambda[1]

  `weight_in_root_lattice`()

    Return the weight of `self` as an element of the root lattice.

    EXAMPLES:

        sage: M = crystals.infinity.NakajimaMonomials(['C',3,1],use_Y=False)
        sage: m = M.module_generators[0].f_string([3,0,1,2,0])
        sage: m.weight_in_root_lattice()
        -2*alpha[0] - alpha[1] - alpha[2] - alpha[3]

class `sage.combinat.crystals.monomial_crystals.``NakajimaYMonomial`(parent, dict)

  Bases: `sage.structure.element.Element`

  Monomials of the form \(Y_{i_1,k_1}^{a_1}\cdots Y_{i_t,k_t}^{y_t}\), where \(i_1,\dots,i_t\) are elements of the index set, \(k_1,\dots,k_t\) are nonnegative integers, and \(y_1,\dots,y_t\) are integers.

  EXAMPLES:

      sage: M = crystals.infinity.NakajimaMonomials(['B',3,1])
      sage: mg = M.module_generators[0]
      sage: mg
      1
      sage: mg.f_string([1,3,2,0,1,2,3,0,0,1])
      Y(0,0)^-1 Y(0,1)^-1 Y(0,2)^-1 Y(0,3)^-1 Y(1,0)^-3 Y(1,1)^-2 Y(1,2) Y(2,0)^3 Y(2,2) Y(3,0) Y(3,2)^-1

  `e`(i)

    Return the action of \(e_i\) on `self`.

    INPUT:

    - `i` – an element of the index set

    EXAMPLES:

        sage: M = crystals.infinity.NakajimaMonomials(['E',7,1])
        sage: m = M.module_generators[0].f_string([0,1,4,3])
        sage: [m.e(i) for i in M.index_set()]
        [None,
         None,
         None,
         Y(0,0)^-1 Y(1,1)^-1 Y(2,1) Y(3,0) Y(3,1) Y(4,0)^-1 Y(4,1)^-1 Y(5,0) ,
         None,
         None,
         None,
         None]

        sage: M = crystals.infinity.NakajimaMonomials("C5")
        sage: m = M.module_generators[0].f_string([1,3])
        sage: [m.e(i) for i in M.index_set()]
        [Y(2,1) Y(3,0)^-1 Y(3,1)^-1 Y(4,0) ,
         None,
         Y(1,0)^-1 Y(1,1)^-1 Y(2,0) ,
         None,
         None]

  `epsilon`(i)

    Return the value of \(\varepsilon_i\) on `self`.

    INPUT:

    - `i` – an element of the index set

    EXAMPLES:

        sage: M = crystals.infinity.NakajimaMonomials(['G',2,1])
        sage: m = M.module_generators[0].f(2)
        sage: [m.epsilon(i) for i in M.index_set()]
        [0, 0, 1]

  `f`(i)

    Return the action of \(f_i\) on `self`.

    INPUT:

    - `i` – an element of the index set

    EXAMPLES:

        sage: M = crystals.infinity.NakajimaMonomials("B4")
        sage: m = M.module_generators[0].f_string([1,3,4])
        sage: [m.f(i) for i in M.index_set()]
        [Y(1,0)^-2 Y(1,1)^-2 Y(2,0)^2 Y(2,1) Y(3,0)^-1 Y(4,0) Y(4,1)^-1 ,
         Y(1,0)^-1 Y(1,1)^-1 Y(1,2) Y(2,0) Y(2,2)^-1 Y(3,0)^-1 Y(3,1) Y(4,0) Y(4,1)^-1 ,
         Y(1,0)^-1 Y(1,1)^-1 Y(2,0) Y(2,1)^2 Y(3,0)^-2 Y(3,1)^-1 Y(4,0)^3 Y(4,1)^-1 ,
         Y(1,0)^-1 Y(1,1)^-1 Y(2,0) Y(2,1) Y(3,0)^-1 Y(3,1) Y(4,1)^-2 ]

  `phi`(i)

    Return the value of \(\varphi_i\) on `self`.

    INPUT:

    - `i` – an element of the index set

    EXAMPLES:

        sage: M = crystals.infinity.NakajimaMonomials(['D',4,3])
        sage: m = M.module_generators[0].f(1)
        sage: [m.phi(i) for i in M.index_set()]
        [1, -1, 1]

  `weight`()

    Return the weight of `self` as an element of `self.parent().weight_lattice_realization`.

    EXAMPLES:

        sage: M = crystals.infinity.NakajimaMonomials(['D',4,2])
        sage: m = M.module_generators[0].f_string([0,3,2,0,1])
        sage: m.weight()
        -2*Lambda[0] + Lambda[1]

        sage: M = crystals.infinity.NakajimaMonomials(['E',6])
        sage: m = M.module_generators[0].f_string([1,5,2,6,3])
        sage: m.weight()
        (-1/2, -3/2, 3/2, 1/2, -1/2, 1/2, 1/2, -1/2)

  `weight_in_root_lattice`()

    Return the weight of `self` as an element of the root lattice.

    EXAMPLES:

        sage: M = crystals.infinity.NakajimaMonomials(['F',4])
        sage: m = M.module_generators[0].f_string([3,3,1,2,4])
        sage: m.weight_in_root_lattice()
        -alpha[1] - alpha[2] - 2*alpha[3] - alpha[4]

        sage: M = crystals.infinity.NakajimaMonomials(['B',3,1])
        sage: mg = M.module_generators[0]
        sage: m = mg.f_string([1,3,2,0,1,2,3,0,0,1])
        sage: m.weight_in_root_lattice()
        -3*alpha[0] - 3*alpha[1] - 2*alpha[2] - 2*alpha[3]

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