Crystals of Modified Nakajima Monomials

AUTHORS:

• Arthur Lubovsky: Initial version

• Ben Salisbury: Initial version

Let $Y_{i,k}$, for $i\in I$ and $k\in \mathbf{Z}$, be a commuting set of variables, and let $\mathbf{1}$ 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 $\hat{\mathcal{M}}$ of monomials of the form

$$M=\prod _{(i,k)\in I\times {\mathbf{Z}}_{\ge 0}}{Y}_{i,k}^{y_i(k)}\mathbf{1}.$$

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

$$\begin{array}{r}\begin{array}{rl}\mathrm{wt}(M)& =\sum _{i\in I}(\sum _{k\ge 0}{y}_i(k)){\mathrm{\Lambda }}_i,\\ {\phi }_i(M)& =max\{\sum _{0\le j\le k}{y}_i(j):k\ge 0\},\\ {\epsilon }_i(M)& ={\phi }_i(M)-⟨h_i,\mathrm{wt}(M)⟩,\\ k_f=k_f(M)& =min\{k\ge 0:{\phi }_i(M)=\sum _{0\le j\le k}{y}_i(j)\},\\ k_e=k_e(M)& =max\{k\ge 0:{\phi }_i(M)=\sum _{0\le j\le k}{y}_i(j)\},\end{array}\end{array}$$

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

$$A_{i,k}=Y_{i,k}{Y}_{i,k+1}\prod _{j\ne i}{Y}_{j,k+c_{ji}}^{a_{ji}},$$

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

$$\begin{array}{r}\begin{array}{rl}{e}_{i}M& =\{\begin{array}{ll}0& \text{if }{\epsilon }_i(M)=0,\\ A_{i,k_e}M& \text{if }{\epsilon }_i(M)>0,\end{array}\\ f_{i}M& =A_{i,k_f}^{-1}M.\end{array}\end{array}$$

It is shown in [KKS2007] that the connected component of $\hat{\mathcal{M}}$ containing the element $\mathbf{1}$, which we denote by $\mathcal{M}(\mathrm{\infty })$, is crystal isomorphic to the crystal $B(\mathrm{\infty })$.

Let $\tilde{\mathcal{M}}$ be $\hat{\mathcal{M}}$ as a set, and with crystal structure defined as on $\hat{\mathcal{M}}$ with the exception that

$$\begin{array}{r}{f}_{i}M=\{\begin{array}{ll}0& \text{if }{\phi }_i(M)=0,\\ A_{i,k_f}^{-1}M& \text{if }{\phi }_i(M)>0.\end{array}\end{array}$$

Then Kashiwara [Ka2003] showed that the connected component in $\tilde{\mathcal{M}}$ containing a monomial $M$ such that $e_{i}M=0$, 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 $C=(c_{ij}{)}_{i\ne j}$ 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)

Bases: InfinityCrystalOfNakajimaMonomials

Let $\tilde{\mathcal{M}}$ be $\hat{\mathcal{M}}$ as a set, and with crystal structure defined as on $\hat{\mathcal{M}}$ with the exception that

$$\begin{array}{r}{f}_{i}M=\{\begin{array}{ll}0& \text{if }{\phi }_i(M)=0,\\ A_{i,k_f}^{-1}M& \text{if }{\phi }_i(M)>0.\end{array}\end{array}$$

Then Kashiwara [Ka2003] showed that the connected component in $\tilde{\mathcal{M}}$ containing a monomial $M$ such that $e_{i}M=0$, 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

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]

Python

>>> 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]

Sage Live

La = RootSystem("A2").weight_lattice().fundamental_weights()
M = crystals.NakajimaMonomials("A2",La[1]+La[2])
B = crystals.Tableaux("A2",shape=[2,1])
GM = M.digraph()
GB = B.digraph()
GM.is_isomorphic(GB,edge_labels=True)
La = RootSystem("G2").weight_lattice().fundamental_weights()
M = crystals.NakajimaMonomials("G2",La[1]+La[2])
B = crystals.Tableaux("G2",shape=[2,1])
GM = M.digraph()
GB = B.digraph()
GM.is_isomorphic(GB,edge_labels=True)
La = RootSystem("B2").weight_lattice().fundamental_weights()
M = crystals.NakajimaMonomials(['B',2],La[1]+La[2])
B = crystals.Tableaux("B2",shape=[3/2,1/2])
GM = M.digraph()
GB = B.digraph()
GM.is_isomorphic(GB,edge_labels=True)
La = RootSystem(['A',3,1]).weight_lattice(extended=True).fundamental_weights()
M = crystals.NakajimaMonomials(['A',3,1],La[0]+La[2])
B = crystals.GeneralizedYoungWalls(3,La[0]+La[2])
SM = M.subcrystal(max_depth=4)
SB = B.subcrystal(max_depth=4)
GM = M.digraph(subset=SM) # long time
GB = B.digraph(subset=SB) # long time
GM.is_isomorphic(GB,edge_labels=True) # long time
La = RootSystem(['A',5,2]).weight_lattice(extended=True).fundamental_weights()
LA = RootSystem(['A',5,2]).weight_space().fundamental_weights()
M = crystals.NakajimaMonomials(['A',5,2],3*La[0])
B = crystals.LSPaths(3*LA[0])
SM = M.subcrystal(max_depth=4)
SB = B.subcrystal(max_depth=4)
GM = M.digraph(subset=SM)
GB = B.digraph(subset=SB)
GM.is_isomorphic(GB,edge_labels=True)
c = matrix([[0,1,0],[0,0,1],[1,0,0]])
La = RootSystem(['A',2,1]).weight_lattice(extended=True).fundamental_weights()
M = crystals.NakajimaMonomials(2*La[1], c=c)
sorted(M.subcrystal(max_depth=3), key=str)

Element

alias of CrystalOfNakajimaMonomialsElement

cardinality()

Return the cardinality of self.

EXAMPLES:

Sage

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

Python

>>> 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

Sage Live

La = RootSystem(['A',2]).weight_lattice().fundamental_weights()
M = crystals.NakajimaMonomials(['A',2], La[1])
M.cardinality()
La = RootSystem(['D',4,2]).weight_lattice(extended=True).fundamental_weights()
M = crystals.NakajimaMonomials(['D',4,2], La[1])
M.cardinality()

class sage.combinat.crystals.monomial_crystals.CrystalOfNakajimaMonomialsElement(parent, Y, A)

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

$$\begin{array}{r}{f}_{i}M=\{\begin{array}{ll}0& \text{if }{\phi }_i(M)=0,\\ A_{i,k_f}^{-1}M& \text{if }{\phi }_i(M)>0.\end{array}\end{array}$$

EXAMPLES:

Sage

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()

Python

>>> 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()

Sage Live

La = RootSystem(['A',5,2]).weight_lattice(extended=True).fundamental_weights()
M = crystals.NakajimaMonomials(['A',5,2],3*La[0])
m = M.module_generators[0].f(0); m
TestSuite(m).run()

f(i)

Return the action of $f_i$ on self.

INPUT:

• i – an element of the index set

EXAMPLES:

Sage

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]

Python

>>> 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 Live

La = RootSystem(['A',5,2]).weight_lattice(extended=True).fundamental_weights()
M = crystals.NakajimaMonomials(['A',5,2],3*La[0])
m = M.module_generators[0]
[m.f(i) for i in M.index_set()]

Sage

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')

Python

>>> 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')

Sage Live

M = crystals.infinity.NakajimaMonomials("E8")
M.set_variables('A')
m = M.module_generators[0].f_string([4,2,3,8])
m
[m.f(i) for i in M.index_set()]
M.set_variables('Y')

weight()

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

EXAMPLES:

Sage

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)

Python

>>> 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)

Sage Live

La = RootSystem("A2").weight_lattice().fundamental_weights()
M = crystals.NakajimaMonomials("A2",La[1]+La[2])
M.module_generators[0].weight()

class sage.combinat.crystals.monomial_crystals.InfinityCrystalOfNakajimaMonomials(ct, c, category=None)

Bases: UniqueRepresentation, Parent

Crystal $B(\mathrm{\infty })$ in terms of (modified) Nakajima monomials.

Let $Y_{i,k}$, for $i\in I$ and $k\in \mathbf{Z}$, be a commuting set of variables, and let $\mathbf{1}$ 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 $\hat{\mathcal{M}}$ of monomials of the form

$$M=\prod _{(i,k)\in I\times {\mathbf{Z}}_{\ge 0}}{Y}_{i,k}^{y_i(k)}\mathbf{1}.$$

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

$$\begin{array}{r}\begin{array}{rl}\mathrm{wt}(M)& =\sum _{i\in I}(\sum _{k\ge 0}{y}_i(k)){\mathrm{\Lambda }}_i,\\ {\phi }_i(M)& =max\{\sum _{0\le j\le k}{y}_i(j):k\ge 0\},\\ {\epsilon }_i(M)& ={\phi }_i(M)-⟨h_i,\mathrm{wt}(M)⟩,\\ k_f=k_f(M)& =min\{k\ge 0:{\phi }_i(M)=\sum _{0\le j\le k}{y}_i(j)\},\\ k_e=k_e(M)& =max\{k\ge 0:{\phi }_i(M)=\sum _{0\le j\le k}{y}_i(j)\},\end{array}\end{array}$$

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

$$A_{i,k}=Y_{i,k}{Y}_{i,k+1}\prod _{j\ne i}{Y}_{j,k+c_{ji}}^{a_{ji}},$$

where $(a_{ij}{)}_{i,j\in I}$ is a Cartan matrix. Then

$$\begin{array}{r}\begin{array}{rl}{e}_{i}M& =\{\begin{array}{ll}0& \text{if }{\epsilon }_i(M)=0,\\ A_{i,k_e}M& \text{if }{\epsilon }_i(M)>0,\end{array}\\ f_{i}M& =A_{i,k_f}^{-1}M.\end{array}\end{array}$$

It is shown in [KKS2007] that the connected component of $\hat{\mathcal{M}}$ containing the element $\mathbf{1}$, which we denote by $\mathcal{M}(\mathrm{\infty })$, is crystal isomorphic to the crystal $B(\mathrm{\infty })$.

INPUT:

• cartan_type – a Cartan type

• c – (optional) the matrix $(c_{ij}{)}_{i,j\in I}$ such that $c_{ii}=0$ for all $i\in I$, $c_{ij}\in {\mathbf{Z}}_{>0}$ for all $i,j\in I$, and $c_{ij}+c_{ji}=1$ for all $i\ne j$; the default is $c_{ij}=0$ if $i<j$ and $0$ otherwise

EXAMPLES:

Sage

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

Python

>>> 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

Sage Live

B = crystals.infinity.Tableaux("C3")
S = B.subcrystal(max_depth=4)
G = B.digraph(subset=S) # long time
M = crystals.infinity.NakajimaMonomials("C3") # long time
T = M.subcrystal(max_depth=4) # long time
H = M.digraph(subset=T) # long time
G.is_isomorphic(H,edge_labels=True) # long time
M = crystals.infinity.NakajimaMonomials(['A',2,1])
T = M.subcrystal(max_depth=3)
H = M.digraph(subset=T) # long time
Y = crystals.infinity.GeneralizedYoungWalls(2)
YS = Y.subcrystal(max_depth=3)
YG = Y.digraph(subset=YS) # long time
YG.is_isomorphic(H,edge_labels=True) # long time
M = crystals.infinity.NakajimaMonomials("D4")
B = crystals.infinity.Tableaux("D4")
MS = M.subcrystal(max_depth=3)
BS = B.subcrystal(max_depth=3)
MG = M.digraph(subset=MS) # long time
BG = B.digraph(subset=BS) # long time
BG.is_isomorphic(MG,edge_labels=True) # long time

Element

alias of NakajimaMonomial

c()

Return the matrix $c_{ij}$ of self.

EXAMPLES:

Sage

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

Python

>>> 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

Sage Live

La = RootSystem(['B',3]).weight_lattice().fundamental_weights()
M = crystals.NakajimaMonomials(La[1]+La[2])
M.c()
c = Matrix([[0,0,1],[1,0,0],[0,1,0]])
La = RootSystem(['A',2,1]).weight_lattice(extended=True).fundamental_weights()
M = crystals.NakajimaMonomials(2*La[1], c=c)
M.c() == c

cardinality()

Return the cardinality of self, which is always $\mathrm{\infty }$.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> M = crystals.infinity.NakajimaMonomials(['A',Integer(5),Integer(2)])
>>> M.cardinality()
+Infinity

Sage Live

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

get_variables()

Return the type of monomials to use for the element output.

EXAMPLES:

Sage

sage: M = crystals.infinity.NakajimaMonomials(['A', 4])
sage: M.get_variables()
'Y'

Python

>>> from sage.all import *
>>> M = crystals.infinity.NakajimaMonomials(['A', Integer(4)])
>>> M.get_variables()
'Y'

Sage Live

M = crystals.infinity.NakajimaMonomials(['A', 4])
M.get_variables()

set_variables(letter)

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 $\sum _{i\in I}{\lambda }_i{\mathrm{\Lambda }}_i$ 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

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')

Python

>>> 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 Live

M = crystals.infinity.NakajimaMonomials(['A', 4])
elt = M.highest_weight_vector().f_string([2,1,3,2,3,2,4,3])
elt
M.set_variables('A')
elt
M.set_variables('Y')

Sage

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')

Python

>>> 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')

Sage Live

La = RootSystem(['A',2]).weight_lattice().fundamental_weights()
M = crystals.NakajimaMonomials(La[1]+La[2])
lw = M.lowest_weight_vectors()[0]
lw
M.set_variables('A')
lw
M.set_variables('Y')

class sage.combinat.crystals.monomial_crystals.NakajimaMonomial(parent, Y, A)

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

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

Python

>>> 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

Sage Live

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

An example using the $A$ variables:

Sage

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')

Python

>>> 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')

Sage Live

M = crystals.infinity.NakajimaMonomials("A3")
M.set_variables('A')
mg = M.module_generators[0]
mg.f_string([1,2,3,2,1])
mg.f_string([3,2,1])
M.set_variables('Y')

e(i)

Return the action of $e_i$ on self.

INPUT:

• i – an element of the index set

EXAMPLES:

Sage

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')

Python

>>> 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')

Sage Live

M = crystals.infinity.NakajimaMonomials(['E',7,1])
m = M.module_generators[0].f_string([0,1,4,3])
[m.e(i) for i in M.index_set()]
M = crystals.infinity.NakajimaMonomials("C5")
m = M.module_generators[0].f_string([1,3])
[m.e(i) for i in M.index_set()]
M = crystals.infinity.NakajimaMonomials(['D',4,1])
M.set_variables('A')
m = M.module_generators[0].f_string([4,2,3,0])
[m.e(i) for i in M.index_set()]
M.set_variables('Y')

epsilon(i)

Return the value of ${\epsilon }_i$ on self.

INPUT:

• i – an element of the index set

EXAMPLES:

Sage

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]

Python

>>> 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]

Sage Live

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

f(i)

Return the action of $f_i$ on self.

INPUT:

• i – an element of the index set

EXAMPLES:

Sage

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]

Python

>>> 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]

Sage Live

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

phi(i)

Return the value of ${\phi }_i$ on self.

INPUT:

• i – an element of the index set

EXAMPLES:

Sage

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]

Python

>>> 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]

Sage Live

M = crystals.infinity.NakajimaMonomials(['D',4,3])
m = M.module_generators[0].f(1)
[m.phi(i) for i in M.index_set()]
M = crystals.infinity.NakajimaMonomials(['C',4,1])
m = M.module_generators[0].f_string([4,2,3])
[m.phi(i) for i in M.index_set()]

weight()

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

EXAMPLES:

Sage

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

Python

>>> 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

Sage Live

C = crystals.infinity.NakajimaMonomials(['A',1,1])
v = C.highest_weight_vector()
v.f(1).weight() + v.f(0).weight()
M = crystals.infinity.NakajimaMonomials(['A',4,2])
m = M.highest_weight_vector().f_string([1,2,0,1])
m.weight()

weight_in_root_lattice()

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

EXAMPLES:

Sage

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]

Python

>>> 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]

Sage Live

M = crystals.infinity.NakajimaMonomials(['F',4])
m = M.module_generators[0].f_string([3,3,1,2,4])
m.weight_in_root_lattice()
M = crystals.infinity.NakajimaMonomials(['B',3,1])
mg = M.module_generators[0]
m = mg.f_string([1,3,2,0,1,2,3,0,0,1])
m.weight_in_root_lattice()
M = crystals.infinity.NakajimaMonomials(['C',3,1])
m = M.module_generators[0].f_string([3,0,1,2,0])
m.weight_in_root_lattice()

Make live