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
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
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
where \((a_{ij})\) is a Cartan matrix. Then
It is shown in [KKS2007] 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
Then Kashiwara [Ka2003] 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 \(i>j\).
- class sage.combinat.crystals.monomial_crystals.CrystalOfNakajimaMonomials(ct, La, c)[source]#
Bases:
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 [Ka2003] 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 typeLa
– 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(extended=True).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(extended=True).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 sage: c = matrix([[0,1,0],[0,0,1],[1,0,0]]) sage: La = RootSystem(['A',2,1]).weight_lattice(extended=True).fundamental_weights() sage: M = crystals.NakajimaMonomials(2*La[1], c=c) sage: sorted(M.subcrystal(max_depth=3), key=str) [Y(0,0) Y(0,1) Y(1,0) Y(2,1)^-1, Y(0,0) Y(0,1)^2 Y(1,1)^-1 Y(2,0) Y(2,1)^-1, Y(0,0) Y(0,2)^-1 Y(1,0) Y(1,1) Y(2,1)^-1 Y(2,2), Y(0,1) Y(0,2)^-1 Y(1,1)^-1 Y(2,0)^2 Y(2,2), Y(0,1) Y(1,0) Y(1,1)^-1 Y(2,0), Y(0,1)^2 Y(1,1)^-2 Y(2,0)^2, Y(0,2)^-1 Y(1,0) Y(2,0) Y(2,2), Y(1,0) Y(1,3) Y(2,0) Y(2,3)^-1, Y(1,0)^2]
>>> from sage.all import * >>> La = RootSystem("A2").weight_lattice().fundamental_weights() >>> M = crystals.NakajimaMonomials("A2",La[Integer(1)]+La[Integer(2)]) >>> B = crystals.Tableaux("A2",shape=[Integer(2),Integer(1)]) >>> GM = M.digraph() >>> GB = B.digraph() >>> GM.is_isomorphic(GB,edge_labels=True) True >>> La = RootSystem("G2").weight_lattice().fundamental_weights() >>> M = crystals.NakajimaMonomials("G2",La[Integer(1)]+La[Integer(2)]) >>> B = crystals.Tableaux("G2",shape=[Integer(2),Integer(1)]) >>> GM = M.digraph() >>> GB = B.digraph() >>> GM.is_isomorphic(GB,edge_labels=True) True >>> La = RootSystem("B2").weight_lattice().fundamental_weights() >>> M = crystals.NakajimaMonomials(['B',Integer(2)],La[Integer(1)]+La[Integer(2)]) >>> B = crystals.Tableaux("B2",shape=[Integer(3)/Integer(2),Integer(1)/Integer(2)]) >>> GM = M.digraph() >>> GB = B.digraph() >>> GM.is_isomorphic(GB,edge_labels=True) True >>> La = RootSystem(['A',Integer(3),Integer(1)]).weight_lattice(extended=True).fundamental_weights() >>> M = crystals.NakajimaMonomials(['A',Integer(3),Integer(1)],La[Integer(0)]+La[Integer(2)]) >>> B = crystals.GeneralizedYoungWalls(Integer(3),La[Integer(0)]+La[Integer(2)]) >>> SM = M.subcrystal(max_depth=Integer(4)) >>> SB = B.subcrystal(max_depth=Integer(4)) >>> GM = M.digraph(subset=SM) # long time >>> GB = B.digraph(subset=SB) # long time >>> GM.is_isomorphic(GB,edge_labels=True) # long time True >>> La = RootSystem(['A',Integer(5),Integer(2)]).weight_lattice(extended=True).fundamental_weights() >>> LA = RootSystem(['A',Integer(5),Integer(2)]).weight_space().fundamental_weights() >>> M = crystals.NakajimaMonomials(['A',Integer(5),Integer(2)],Integer(3)*La[Integer(0)]) >>> B = crystals.LSPaths(Integer(3)*LA[Integer(0)]) >>> SM = M.subcrystal(max_depth=Integer(4)) >>> SB = B.subcrystal(max_depth=Integer(4)) >>> GM = M.digraph(subset=SM) >>> GB = B.digraph(subset=SB) >>> GM.is_isomorphic(GB,edge_labels=True) True >>> c = matrix([[Integer(0),Integer(1),Integer(0)],[Integer(0),Integer(0),Integer(1)],[Integer(1),Integer(0),Integer(0)]]) >>> La = RootSystem(['A',Integer(2),Integer(1)]).weight_lattice(extended=True).fundamental_weights() >>> M = crystals.NakajimaMonomials(Integer(2)*La[Integer(1)], c=c) >>> sorted(M.subcrystal(max_depth=Integer(3)), key=str) [Y(0,0) Y(0,1) Y(1,0) Y(2,1)^-1, Y(0,0) Y(0,1)^2 Y(1,1)^-1 Y(2,0) Y(2,1)^-1, Y(0,0) Y(0,2)^-1 Y(1,0) Y(1,1) Y(2,1)^-1 Y(2,2), Y(0,1) Y(0,2)^-1 Y(1,1)^-1 Y(2,0)^2 Y(2,2), Y(0,1) Y(1,0) Y(1,1)^-1 Y(2,0), Y(0,1)^2 Y(1,1)^-2 Y(2,0)^2, Y(0,2)^-1 Y(1,0) Y(2,0) Y(2,2), Y(1,0) Y(1,3) Y(2,0) Y(2,3)^-1, Y(1,0)^2]
- Element[source]#
alias of
CrystalOfNakajimaMonomialsElement
- cardinality()[source]#
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(extended=True).fundamental_weights() sage: M = crystals.NakajimaMonomials(['D',4,2], La[1]) sage: M.cardinality() +Infinity
>>> from sage.all import * >>> La = RootSystem(['A',Integer(2)]).weight_lattice().fundamental_weights() >>> M = crystals.NakajimaMonomials(['A',Integer(2)], La[Integer(1)]) >>> M.cardinality() 3 >>> La = RootSystem(['D',Integer(4),Integer(2)]).weight_lattice(extended=True).fundamental_weights() >>> M = crystals.NakajimaMonomials(['D',Integer(4),Integer(2)], La[Integer(1)]) >>> M.cardinality() +Infinity
- class sage.combinat.crystals.monomial_crystals.CrystalOfNakajimaMonomialsElement(parent, Y, A)[source]#
Bases:
NakajimaMonomial
Element class for
CrystalOfNakajimaMonomials
.The \(f_i\) operators need to be modified from the version in
monomial_crystalsNakajimaMonomial
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(extended=True).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()
>>> from sage.all import * >>> La = RootSystem(['A',Integer(5),Integer(2)]).weight_lattice(extended=True).fundamental_weights() >>> M = crystals.NakajimaMonomials(['A',Integer(5),Integer(2)],Integer(3)*La[Integer(0)]) >>> m = M.module_generators[Integer(0)].f(Integer(0)); m Y(0,0)^2 Y(0,1)^-1 Y(2,0) >>> TestSuite(m).run()
- f(i)[source]#
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(extended=True).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]
>>> from sage.all import * >>> La = RootSystem(['A',Integer(5),Integer(2)]).weight_lattice(extended=True).fundamental_weights() >>> M = crystals.NakajimaMonomials(['A',Integer(5),Integer(2)],Integer(3)*La[Integer(0)]) >>> m = M.module_generators[Integer(0)] >>> [m.f(i) for i in M.index_set()] [Y(0,0)^2 Y(0,1)^-1 Y(2,0), None, None, None]
sage: M = crystals.infinity.NakajimaMonomials("E8") sage: M.set_variables('A') 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] sage: M.set_variables('Y')
>>> from sage.all import * >>> M = crystals.infinity.NakajimaMonomials("E8") >>> M.set_variables('A') >>> m = M.module_generators[Integer(0)].f_string([Integer(4),Integer(2),Integer(3),Integer(8)]) >>> m A(2,1)^-1 A(3,1)^-1 A(4,0)^-1 A(8,0)^-1 >>> [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] >>> M.set_variables('Y')
- weight()[source]#
Return the weight of
self
as 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: M.module_generators[0].weight() (2, 1, 0)
>>> from sage.all import * >>> La = RootSystem("A2").weight_lattice().fundamental_weights() >>> M = crystals.NakajimaMonomials("A2",La[Integer(1)]+La[Integer(2)]) >>> M.module_generators[Integer(0)].weight() (2, 1, 0)
- class sage.combinat.crystals.monomial_crystals.InfinityCrystalOfNakajimaMonomials(ct, c, category=None)[source]#
Bases:
UniqueRepresentation
,Parent
Crystal \(B(\infty)\) in terms of (modified) Nakajima monomials.
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 non-negative 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 \((a_{ij})_{i,j \in I}\) 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 [KKS2007] 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 typec
– (optional) the matrix \((c_{ij})_{i,j \in I}\) such that \(c_{ii} = 0\) for all \(i \in I\), \(c_{ij} \in \ZZ_{>0}\) for all \(i,j \in I\), and \(c_{ij} + c_{ji} = 1\) for all \(i \neq j\); the default is \(c_{ij} = 0\) if \(i < j\) and \(0\) otherwise
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
>>> from sage.all import * >>> B = crystals.infinity.Tableaux("C3") >>> S = B.subcrystal(max_depth=Integer(4)) >>> G = B.digraph(subset=S) # long time >>> M = crystals.infinity.NakajimaMonomials("C3") # long time >>> T = M.subcrystal(max_depth=Integer(4)) # long time >>> H = M.digraph(subset=T) # long time >>> G.is_isomorphic(H,edge_labels=True) # long time True >>> M = crystals.infinity.NakajimaMonomials(['A',Integer(2),Integer(1)]) >>> T = M.subcrystal(max_depth=Integer(3)) >>> H = M.digraph(subset=T) # long time >>> Y = crystals.infinity.GeneralizedYoungWalls(Integer(2)) >>> YS = Y.subcrystal(max_depth=Integer(3)) >>> YG = Y.digraph(subset=YS) # long time >>> YG.is_isomorphic(H,edge_labels=True) # long time True >>> M = crystals.infinity.NakajimaMonomials("D4") >>> B = crystals.infinity.Tableaux("D4") >>> MS = M.subcrystal(max_depth=Integer(3)) >>> BS = B.subcrystal(max_depth=Integer(3)) >>> MG = M.digraph(subset=MS) # long time >>> BG = B.digraph(subset=BS) # long time >>> BG.is_isomorphic(MG,edge_labels=True) # long time True
- Element[source]#
alias of
NakajimaMonomial
- c()[source]#
Return the matrix \(c_{ij}\) of
self
.EXAMPLES:
sage: La = RootSystem(['B',3]).weight_lattice().fundamental_weights() sage: M = crystals.NakajimaMonomials(La[1]+La[2]) sage: M.c() [0 1 1] [0 0 1] [0 0 0] sage: c = Matrix([[0,0,1],[1,0,0],[0,1,0]]) sage: La = RootSystem(['A',2,1]).weight_lattice(extended=True).fundamental_weights() sage: M = crystals.NakajimaMonomials(2*La[1], c=c) sage: M.c() == c True
>>> from sage.all import * >>> La = RootSystem(['B',Integer(3)]).weight_lattice().fundamental_weights() >>> M = crystals.NakajimaMonomials(La[Integer(1)]+La[Integer(2)]) >>> M.c() [0 1 1] [0 0 1] [0 0 0] >>> c = Matrix([[Integer(0),Integer(0),Integer(1)],[Integer(1),Integer(0),Integer(0)],[Integer(0),Integer(1),Integer(0)]]) >>> La = RootSystem(['A',Integer(2),Integer(1)]).weight_lattice(extended=True).fundamental_weights() >>> M = crystals.NakajimaMonomials(Integer(2)*La[Integer(1)], c=c) >>> M.c() == c True
- cardinality()[source]#
Return the cardinality of
self
, which is always \(\infty\).EXAMPLES:
sage: M = crystals.infinity.NakajimaMonomials(['A',5,2]) sage: M.cardinality() +Infinity
>>> from sage.all import * >>> M = crystals.infinity.NakajimaMonomials(['A',Integer(5),Integer(2)]) >>> M.cardinality() +Infinity
- get_variables()[source]#
Return the type of monomials to use for the element output.
EXAMPLES:
sage: M = crystals.infinity.NakajimaMonomials(['A', 4]) sage: M.get_variables() 'Y'
>>> from sage.all import * >>> M = crystals.infinity.NakajimaMonomials(['A', Integer(4)]) >>> M.get_variables() 'Y'
- set_variables(letter)[source]#
Set the type of monomials to use for the element output.
If the \(A\) variables are used, the output is written as \(\prod_{i\in I} Y_{i,0}^{\lambda_i} \prod_{i,k} A_{i,k}^{c_{i,k}}\), where \(\sum_{i \in I} \lambda_i \Lambda_i\) is the corresponding dominant weight.
INPUT:
letter
– can be one of the following:'Y'
– use \(Y_{i,k}\), corresponds to fundamental weights'A'
– use \(A_{i,k}\), corresponds to simple roots
EXAMPLES:
sage: M = crystals.infinity.NakajimaMonomials(['A', 4]) sage: elt = M.highest_weight_vector().f_string([2,1,3,2,3,2,4,3]) sage: elt Y(1,2) Y(2,0)^-1 Y(2,2)^-1 Y(3,0)^-1 Y(3,2)^-1 Y(4,0) sage: M.set_variables('A') sage: elt A(1,1)^-1 A(2,0)^-1 A(2,1)^-2 A(3,0)^-2 A(3,1)^-1 A(4,0)^-1 sage: M.set_variables('Y')
>>> from sage.all import * >>> M = crystals.infinity.NakajimaMonomials(['A', Integer(4)]) >>> elt = M.highest_weight_vector().f_string([Integer(2),Integer(1),Integer(3),Integer(2),Integer(3),Integer(2),Integer(4),Integer(3)]) >>> elt Y(1,2) Y(2,0)^-1 Y(2,2)^-1 Y(3,0)^-1 Y(3,2)^-1 Y(4,0) >>> M.set_variables('A') >>> elt A(1,1)^-1 A(2,0)^-1 A(2,1)^-2 A(3,0)^-2 A(3,1)^-1 A(4,0)^-1 >>> M.set_variables('Y')
sage: La = RootSystem(['A',2]).weight_lattice().fundamental_weights() sage: M = crystals.NakajimaMonomials(La[1]+La[2]) sage: lw = M.lowest_weight_vectors()[0] sage: lw Y(1,2)^-1 Y(2,1)^-1 sage: M.set_variables('A') sage: lw Y(1,0) Y(2,0) A(1,0)^-1 A(1,1)^-1 A(2,0)^-2 sage: M.set_variables('Y')
>>> from sage.all import * >>> La = RootSystem(['A',Integer(2)]).weight_lattice().fundamental_weights() >>> M = crystals.NakajimaMonomials(La[Integer(1)]+La[Integer(2)]) >>> lw = M.lowest_weight_vectors()[Integer(0)] >>> lw Y(1,2)^-1 Y(2,1)^-1 >>> M.set_variables('A') >>> lw Y(1,0) Y(2,0) A(1,0)^-1 A(1,1)^-1 A(2,0)^-2 >>> M.set_variables('Y')
- class sage.combinat.crystals.monomial_crystals.NakajimaMonomial(parent, Y, A)[source]#
Bases:
Element
An element of the monomial crystal.
Monomials of the form \(Y_{i_1,k_1}^{y_1} \cdots Y_{i_t,k_t}^{y_t}\), 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
>>> from sage.all import * >>> M = crystals.infinity.NakajimaMonomials(['B',Integer(3),Integer(1)]) >>> mg = M.module_generators[Integer(0)] >>> mg 1 >>> mg.f_string([Integer(1),Integer(3),Integer(2),Integer(0),Integer(1),Integer(2),Integer(3),Integer(0),Integer(0),Integer(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
An example using the \(A\) variables:
sage: M = crystals.infinity.NakajimaMonomials("A3") sage: M.set_variables('A') 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 sage: M.set_variables('Y')
>>> from sage.all import * >>> M = crystals.infinity.NakajimaMonomials("A3") >>> M.set_variables('A') >>> mg = M.module_generators[Integer(0)] >>> mg.f_string([Integer(1),Integer(2),Integer(3),Integer(2),Integer(1)]) A(1,0)^-1 A(1,1)^-1 A(2,0)^-2 A(3,0)^-1 >>> mg.f_string([Integer(3),Integer(2),Integer(1)]) A(1,2)^-1 A(2,1)^-1 A(3,0)^-1 >>> M.set_variables('Y')
- e(i)[source]#
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] sage: M = crystals.infinity.NakajimaMonomials(['D',4,1]) sage: M.set_variables('A') 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] sage: M.set_variables('Y')
>>> from sage.all import * >>> M = crystals.infinity.NakajimaMonomials(['E',Integer(7),Integer(1)]) >>> m = M.module_generators[Integer(0)].f_string([Integer(0),Integer(1),Integer(4),Integer(3)]) >>> [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] >>> M = crystals.infinity.NakajimaMonomials("C5") >>> m = M.module_generators[Integer(0)].f_string([Integer(1),Integer(3)]) >>> [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] >>> M = crystals.infinity.NakajimaMonomials(['D',Integer(4),Integer(1)]) >>> M.set_variables('A') >>> m = M.module_generators[Integer(0)].f_string([Integer(4),Integer(2),Integer(3),Integer(0)]) >>> [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] >>> M.set_variables('Y')
- epsilon(i)[source]#
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] sage: M = crystals.infinity.NakajimaMonomials(['C',4,1]) 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]
>>> from sage.all import * >>> M = crystals.infinity.NakajimaMonomials(['G',Integer(2),Integer(1)]) >>> m = M.module_generators[Integer(0)].f(Integer(2)) >>> [m.epsilon(i) for i in M.index_set()] [0, 0, 1] >>> M = crystals.infinity.NakajimaMonomials(['C',Integer(4),Integer(1)]) >>> m = M.module_generators[Integer(0)].f_string([Integer(4),Integer(2),Integer(3)]) >>> [m.epsilon(i) for i in M.index_set()] [0, 0, 0, 1, 0]
- f(i)[source]#
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]
>>> from sage.all import * >>> M = crystals.infinity.NakajimaMonomials("B4") >>> m = M.module_generators[Integer(0)].f_string([Integer(1),Integer(3),Integer(4)]) >>> [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)[source]#
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] sage: M = crystals.infinity.NakajimaMonomials(['C',4,1]) 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]
>>> from sage.all import * >>> M = crystals.infinity.NakajimaMonomials(['D',Integer(4),Integer(3)]) >>> m = M.module_generators[Integer(0)].f(Integer(1)) >>> [m.phi(i) for i in M.index_set()] [1, -1, 1] >>> M = crystals.infinity.NakajimaMonomials(['C',Integer(4),Integer(1)]) >>> m = M.module_generators[Integer(0)].f_string([Integer(4),Integer(2),Integer(3)]) >>> [m.phi(i) for i in M.index_set()] [0, 1, -1, 2, -1]
- weight()[source]#
Return the weight of
self
as an element of the weight lattice.EXAMPLES:
sage: C = crystals.infinity.NakajimaMonomials(['A',1,1]) sage: v = C.highest_weight_vector() sage: v.f(1).weight() + v.f(0).weight() -delta sage: M = crystals.infinity.NakajimaMonomials(['A',4,2]) sage: m = M.highest_weight_vector().f_string([1,2,0,1]) sage: m.weight() 2*Lambda[0] - Lambda[1] - delta
>>> from sage.all import * >>> C = crystals.infinity.NakajimaMonomials(['A',Integer(1),Integer(1)]) >>> v = C.highest_weight_vector() >>> v.f(Integer(1)).weight() + v.f(Integer(0)).weight() -delta >>> M = crystals.infinity.NakajimaMonomials(['A',Integer(4),Integer(2)]) >>> m = M.highest_weight_vector().f_string([Integer(1),Integer(2),Integer(0),Integer(1)]) >>> m.weight() 2*Lambda[0] - Lambda[1] - delta
- weight_in_root_lattice()[source]#
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] sage: M = crystals.infinity.NakajimaMonomials(['C',3,1]) 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]
>>> from sage.all import * >>> M = crystals.infinity.NakajimaMonomials(['F',Integer(4)]) >>> m = M.module_generators[Integer(0)].f_string([Integer(3),Integer(3),Integer(1),Integer(2),Integer(4)]) >>> m.weight_in_root_lattice() -alpha[1] - alpha[2] - 2*alpha[3] - alpha[4] >>> M = crystals.infinity.NakajimaMonomials(['B',Integer(3),Integer(1)]) >>> mg = M.module_generators[Integer(0)] >>> m = mg.f_string([Integer(1),Integer(3),Integer(2),Integer(0),Integer(1),Integer(2),Integer(3),Integer(0),Integer(0),Integer(1)]) >>> m.weight_in_root_lattice() -3*alpha[0] - 3*alpha[1] - 2*alpha[2] - 2*alpha[3] >>> M = crystals.infinity.NakajimaMonomials(['C',Integer(3),Integer(1)]) >>> m = M.module_generators[Integer(0)].f_string([Integer(3),Integer(0),Integer(1),Integer(2),Integer(0)]) >>> m.weight_in_root_lattice() -2*alpha[0] - alpha[1] - alpha[2] - alpha[3]