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# Littelmann paths

AUTHORS:

- Mark Shimozono, Anne Schilling (2012): Initial version
- Anne Schilling (2013): Implemented `CrystalOfProjectedLevelZeroLSPaths`

class `sage.combinat.crystals.littelmann_path.``CrystalOfLSPaths`(starting_weight)

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

  Crystal graph of LS paths generated from the straight-line path to a given weight.

  INPUT:

  - `cartan_type` – (optional) the Cartan type of a finite or affine root system
  - `starting_weight` – a weight; if `cartan_type` is given, then the weight should be given as a list of coefficients of the fundamental weights, otherwise it should be given in the `weight_space` basis; for affine highest weight crystals, one needs to use the extended weight space.

  The crystal class of piecewise linear paths in the weight space, generated from a straight-line path from the origin to a given element of the weight lattice.

  OUTPUT:

  - a tuple of weights defining the directions of the piecewise linear segments

  EXAMPLES:

      sage: R = RootSystem(['A',2,1])
      sage: La = R.weight_space(extended = True).basis()
      sage: B = crystals.LSPaths(La[2]-La[0]); B
      The crystal of LS paths of type ['A', 2, 1] and weight -Lambda[0] + Lambda[2]

      sage: C = crystals.LSPaths(['A',2,1],[-1,0,1]); C
      The crystal of LS paths of type ['A', 2, 1] and weight -Lambda[0] + Lambda[2]
      sage: B == C
      True
      sage: c = C.module_generators[0]; c
      (-Lambda[0] + Lambda[2],)
      sage: [c.f(i) for i in C.index_set()]
      [None, None, (Lambda[1] - Lambda[2],)]

      sage: R = C.R; R
      Root system of type ['A', 2, 1]
      sage: Lambda = R.weight_space().basis(); Lambda
      Finite family {0: Lambda[0], 1: Lambda[1], 2: Lambda[2]}
      sage: b=C(tuple([-Lambda[0]+Lambda[2]]))
      sage: b==c
      True
      sage: b.f(2)
      (Lambda[1] - Lambda[2],)

  For classical highest weight crystals we can also compare the results with the tableaux implementation:

      sage: C = crystals.LSPaths(['A',2],[1,1])
      sage: sorted(C, key=str)
      [(-2*Lambda[1] + Lambda[2],), (-Lambda[1] + 1/2*Lambda[2], Lambda[1] - 1/2*Lambda[2]),
       (-Lambda[1] + 2*Lambda[2],), (-Lambda[1] - Lambda[2],),
       (1/2*Lambda[1] - Lambda[2], -1/2*Lambda[1] + Lambda[2]), (2*Lambda[1] - Lambda[2],),
       (Lambda[1] + Lambda[2],), (Lambda[1] - 2*Lambda[2],)]
      sage: C.cardinality()
      8
      sage: B = crystals.Tableaux(['A',2],shape=[2,1])
      sage: B.cardinality()
      8
      sage: B.digraph().is_isomorphic(C.digraph())
      True

  Make sure you use the weight space and not the weight lattice for your weights:

      sage: R = RootSystem(['A',2,1])
      sage: La = R.weight_lattice(extended = True).basis()
      sage: B = crystals.LSPaths(La[2]); B
      Traceback (most recent call last):
      ...
      ValueError: Please use the weight space, rather than weight lattice for your weights

  REFERENCES:

  [Littelmann95] | P. Littelmann, Paths and root operators in representation theory. Ann. of Math. (2) 142 (1995), no. 3, 499-525.

  class `Element`

    Bases: `sage.structure.element_wrapper.ElementWrapper`

    TESTS:

        sage: C = crystals.LSPaths(['E',6],[1,0,0,0,0,0])
        sage: c=C.an_element()
        sage: TestSuite(c).run()

    `compress`()

      Merges consecutive positively parallel steps present in the path.

      EXAMPLES:

          sage: C = crystals.LSPaths(['A',2],[1,1])
          sage: Lambda = C.R.weight_space().fundamental_weights(); Lambda
          Finite family {1: Lambda[1], 2: Lambda[2]}
          sage: c = C(tuple([1/2*Lambda[1]+1/2*Lambda[2], 1/2*Lambda[1]+1/2*Lambda[2]]))
          sage: c.compress()
          (Lambda[1] + Lambda[2],)

    `dualize`()

      Returns dualized path.

      EXAMPLES:

          sage: C = crystals.LSPaths(['A',2],[1,1])
          sage: for c in C:
          ...     print c, c.dualize()
          ...
          (Lambda[1] + Lambda[2],) (-Lambda[1] - Lambda[2],)
          (-Lambda[1] + 2*Lambda[2],) (Lambda[1] - 2*Lambda[2],)
          (1/2*Lambda[1] - Lambda[2], -1/2*Lambda[1] + Lambda[2]) (1/2*Lambda[1] - Lambda[2], -1/2*Lambda[1] + Lambda[2])
          (Lambda[1] - 2*Lambda[2],) (-Lambda[1] + 2*Lambda[2],)
          (-Lambda[1] - Lambda[2],) (Lambda[1] + Lambda[2],)
          (2*Lambda[1] - Lambda[2],) (-2*Lambda[1] + Lambda[2],)
          (-Lambda[1] + 1/2*Lambda[2], Lambda[1] - 1/2*Lambda[2]) (-Lambda[1] + 1/2*Lambda[2], Lambda[1] - 1/2*Lambda[2])
          (-2*Lambda[1] + Lambda[2],) (2*Lambda[1] - Lambda[2],)

    `e`(i, power=1, to_string_end=False, length_only=False)

      Returns the \(i\)-th crystal raising operator on `self`.

      INPUT:

      - `i` – element of the index set of the underlying root system
      - `power` – positive integer; specifies the power of the raising operator to be applied (default: 1)
      - `to_string_end` – boolean; if set to True, returns the dominant end of the \(i\)-string of `self`. (default: False)
      - `length_only` – boolean; if set to True, returns the distance to the dominant end of the \(i\)-string of `self`.

      EXAMPLES:

          sage: C = crystals.LSPaths(['A',2],[1,1])
          sage: c = C[2]; c
          (1/2*Lambda[1] - Lambda[2], -1/2*Lambda[1] + Lambda[2])
          sage: c.e(1)
          sage: c.e(2)
          (-Lambda[1] + 2*Lambda[2],)
          sage: c.e(2,to_string_end=True)
          (-Lambda[1] + 2*Lambda[2],)
          sage: c.e(1,to_string_end=True)
          (1/2*Lambda[1] - Lambda[2], -1/2*Lambda[1] + Lambda[2])
          sage: c.e(1,length_only=True)
          0

    `endpoint`()

      Computes the endpoint of the path.

      EXAMPLES:

          sage: C = crystals.LSPaths(['A',2],[1,1])
          sage: b = C.module_generators[0]
          sage: b.endpoint()
          Lambda[1] + Lambda[2]
          sage: b.f_string([1,2,2,1])
          (-Lambda[1] - Lambda[2],)
          sage: b.f_string([1,2,2,1]).endpoint()
          -Lambda[1] - Lambda[2]
          sage: b.f_string([1,2])
          (1/2*Lambda[1] - Lambda[2], -1/2*Lambda[1] + Lambda[2])
          sage: b.f_string([1,2]).endpoint()
          0
          sage: b = C([])
          sage: b.endpoint()
          0

    `epsilon`(i)

      Returns the distance to the beginning of the \(i\)-string.

      This method overrides the generic implementation in the category of crystals since this computation is more efficient.

      EXAMPLES:

          sage: C = crystals.LSPaths(['A',2],[1,1])
          sage: [c.epsilon(1) for c in C]
          [0, 1, 0, 0, 1, 0, 1, 2]
          sage: [c.epsilon(2) for c in C]
          [0, 0, 1, 2, 1, 1, 0, 0]

    `f`(i, power=1, to_string_end=False, length_only=False)

      Returns the \(i\)-th crystal lowering operator on `self`.

      INPUT:

      - `i` – element of the index set of the underlying root system
      - `power` – positive integer; specifies the power of the lowering operator to be applied (default: 1)
      - `to_string_end` – boolean; if set to True, returns the anti-dominant end of the \(i\)-string of `self`. (default: False)
      - `length_only` – boolean; if set to True, returns the distance to the anti-dominant end of the \(i\)-string of `self`.

      EXAMPLES:

          sage: C = crystals.LSPaths(['A',2],[1,1])
          sage: c = C.module_generators[0]
          sage: c.f(1)
          (-Lambda[1] + 2*Lambda[2],)
          sage: c.f(1,power=2)
          sage: c.f(2)
          (2*Lambda[1] - Lambda[2],)
          sage: c.f(2,to_string_end=True)
          (2*Lambda[1] - Lambda[2],)
          sage: c.f(2,length_only=True)
          1

          sage: C = crystals.LSPaths(['A',2,1],[-1,-1,2])
          sage: c = C.module_generators[0]
          sage: c.f(2,power=2)
          (Lambda[0] + Lambda[1] - 2*Lambda[2],)

    `phi`(i)

      Returns the distance to the end of the \(i\)-string.

      This method overrides the generic implementation in the category of crystals since this computation is more efficient.

      EXAMPLES:

          sage: C = crystals.LSPaths(['A',2],[1,1])
          sage: [c.phi(1) for c in C]
          [1, 0, 0, 1, 0, 2, 1, 0]
          sage: [c.phi(2) for c in C]
          [1, 2, 1, 0, 0, 0, 0, 1]

    `reflect_step`(which_step, i)

      Apply the \(i\)-th simple reflection to the indicated step in `self`.

      EXAMPLES:

          sage: C = crystals.LSPaths(['A',2],[1,1])
          sage: b = C.module_generators[0]
          sage: b.reflect_step(0,1)
          (-Lambda[1] + 2*Lambda[2],)
          sage: b.reflect_step(0,2)
          (2*Lambda[1] - Lambda[2],)

    `s`(i)

      Computes the reflection of `self` along the \(i\)-string.

      This method is more efficient than the generic implementation since it uses powers of \(e\) and \(f\) in the Littelmann model directly.

      EXAMPLES:

          sage: C = crystals.LSPaths(['A',2],[1,1])
          sage: c = C.module_generators[0]
          sage: c.s(1)
          (-Lambda[1] + 2*Lambda[2],)
          sage: c.s(2)
          (2*Lambda[1] - Lambda[2],)

          sage: C = crystals.LSPaths(['A',2,1],[-1,0,1])
          sage: c = C.module_generators[0]; c
          (-Lambda[0] + Lambda[2],)
          sage: c.s(2)
          (Lambda[1] - Lambda[2],)
          sage: c.s(1)
          (-Lambda[0] + Lambda[2],)
          sage: c.f(2).s(1)
          (Lambda[0] - Lambda[1],)

    `split_step`(which_step, r)

      Splits indicated step into two parallel steps of relative lengths \(r\) and \(1-r\).

      INPUT:

      - `which_step` – a position in the tuple `self`
      - `r` – a rational number between 0 and 1

      EXAMPLES:

          sage: C = crystals.LSPaths(['A',2],[1,1])
          sage: b = C.module_generators[0]
          sage: b.split_step(0,1/3)
          (1/3*Lambda[1] + 1/3*Lambda[2], 2/3*Lambda[1] + 2/3*Lambda[2])

class `sage.combinat.crystals.littelmann_path.``CrystalOfProjectedLevelZeroLSPaths`(starting_weight)

  Bases: `sage.combinat.crystals.littelmann_path.CrystalOfLSPaths`

  Crystal of projected level zero LS paths.

  INPUT:

  - `weight` – a dominant weight of the weight space of an affine Kac-Moody root system

  When `weight` is just a single fundamental weight \(\Lambda_r\), this crystal is isomorphic to a Kirillov-Reshetikhin (KR) crystal, see also `sage.combinat.crystals.kirillov_reshetikhin.crystals.KirillovReshetikhinFromLSPaths()`. For general weights, it is isomorphic to a tensor product of single-column KR crystals.

  EXAMPLES:

      sage: R = RootSystem(['C',3,1])
      sage: La = R.weight_space().basis()
      sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[3])
      sage: LS.cardinality()
      84
      sage: GLS = LS.digraph()

      sage: K1 = crystals.KirillovReshetikhin(['C',3,1],1,1)
      sage: K3 = crystals.KirillovReshetikhin(['C',3,1],3,1)
      sage: T = crystals.TensorProduct(K3,K1)
      sage: T.cardinality()
      84
      sage: GT = T.digraph() # long time
      sage: GLS.is_isomorphic(GT, edge_labels = True) # long time
      True

  TESTS:

      sage: ct = CartanType(['A',4,2]).dual()
      sage: P = RootSystem(ct).weight_space()
      sage: La = P.fundamental_weights()
      sage: C = crystals.ProjectedLevelZeroLSPaths(La[1])
      sage: sorted(C, key=str)
      [(-Lambda[0] + Lambda[1],),
       (-Lambda[1] + 2*Lambda[2],),
       (1/2*Lambda[1] - Lambda[2], -1/2*Lambda[1] + Lambda[2]),
       (Lambda[0] - Lambda[1],),
       (Lambda[1] - 2*Lambda[2],)]

  class `Element`

    Bases: `sage.combinat.crystals.littelmann_path.CrystalOfLSPaths.Element`

    Element of a crystal of projected level zero LS paths.

    `energy_function`()

      Return the energy function of `self`.

      The energy function \(D(\pi)\) of the level zero LS path \(\pi \in \mathbb{B}_\mathrm{cl}(\lambda)\) requires a series of definitions; for simplicity the root system is assumed to be untwisted affine.

      The LS path \(\pi\) is a piecewise linear map from the unit interval \([0,1]\) to the weight lattice. It is specified by “times” \(0=\sigma_0<\sigma_1<\dotsm<\sigma_s=1\) and “direction vectors” \(x_u \lambda\) where \(x_u \in W/W_J\) for \(1\le u\le s\), and \(W_J\) is the stabilizer of \(\lambda\) in the finite Weyl group \(W\). Precisely,

      \[\pi(t)=\sum_{u'=1}^{u-1} (\sigma_{u'}-\sigma_{u'-1})x_{u'}\lambda+(t-\sigma_{u-1})x_{u}\lambda\]

      for \(1\le u\le s\) and \(\sigma_{u-1} \le t \le \sigma_{u}\).

      For any \(x,y\in W/W_J\) let

      \[d: x= w_{0} \stackrel{\beta_{1}}{\leftarrow} w_{1} \stackrel{\beta_{2}}{\leftarrow} \cdots \stackrel{\beta_{n}}{\leftarrow} w_{n}=y\]

      be a shortest directed path in the parabolic quantum Bruhat graph. Define

      \[\begin{split}\mathrm{wt}(d):=\sum_{\substack{1\le k\le n \\ \ell(w_{k-1})<\ell(w_k)}} \beta_{k}^{\vee}\end{split}\]

      It can be shown that \(\mathrm{wt}(d)\) depends only on \(x,y\); call its value \(\mathrm{wt}(x,y)\). The energy function \(D(\pi)\) is defined by

      \[D(\pi)=-\sum_{u=1}^{s-1} (1-\sigma_{u}) \langle \lambda,\mathrm{wt}(x_u,x_{u+1}) \rangle\]

      For more information, see [LNSSS2013].

      REFERENCES:

      [LNSSS2013] | (1, 2) C. Lenart, S. Naito, D. Sagaki, A. Schilling, M. Shimozono, A uniform model for Kirillov-Reshetikhin crystals. Extended abstract. DMTCS proc, to appear ( {{{Arxiv 1211.6019}}} )

      Note

      In the dual-of-untwisted case the parabolic quantum Bruhat graph that is used is obtained by exchanging the roles of roots and coroots. Moreover, in the computation of the pairing the short roots must be doubled (or tripled for type \(G\)). This factor is determined by the translation factor of the corresponding root. Type \(BC\) is viewed as untwisted type, whereas the dual of \(BC\) is viewed as twisted. Except for the untwisted cases, these formulas are currently still conjectural.

      EXAMPLES:

          sage: R = RootSystem(['C',3,1])
          sage: La = R.weight_space().basis()
          sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[3])
          sage: b = LS.module_generators[0]
          sage: c = b.f(1).f(3).f(2)
          sage: c.energy_function()
          0
          sage: c=b.e(0)
          sage: c.energy_function()
          1

          sage: R = RootSystem(['A',2,1])
          sage: La = R.weight_space().basis()
          sage: LS = crystals.ProjectedLevelZeroLSPaths(2*La[1])
          sage: b = LS.module_generators[0]
          sage: c = b.e(0)
          sage: c.energy_function()
          1
          sage: for c in sorted(LS, key=str): print c,c.energy_function()
          (-2*Lambda[0] + 2*Lambda[1],)                    0
          (-2*Lambda[1] + 2*Lambda[2],)                    0
          (-Lambda[0] + Lambda[1], -Lambda[1] + Lambda[2]) 1
          (-Lambda[0] + Lambda[1], Lambda[0] - Lambda[2])  1
          (-Lambda[1] + Lambda[2], -Lambda[0] + Lambda[1]) 0
          (-Lambda[1] + Lambda[2], Lambda[0] - Lambda[2])  1
          (2*Lambda[0] - 2*Lambda[2],)                     0
          (Lambda[0] - Lambda[2], -Lambda[0] + Lambda[1])  0
          (Lambda[0] - Lambda[2], -Lambda[1] + Lambda[2])  0

      The next test checks that the energy function is constant on classically connected components:

          sage: R = RootSystem(['A',2,1])
          sage: La = R.weight_space().basis()
          sage: LS = crystals.ProjectedLevelZeroLSPaths(2*La[1]+La[2])
          sage: G = LS.digraph(index_set=[1,2])
          sage: C = G.connected_components()
          sage: [all(c[0].energy_function()==a.energy_function() for a in c) for c in C]
          [True, True, True, True]

          sage: R = RootSystem(['D',4,2])
          sage: La = R.weight_space().basis()
          sage: LS = crystals.ProjectedLevelZeroLSPaths(La[2])
          sage: J = R.cartan_type().classical().index_set()
          sage: hw = [x for x in LS if x.is_highest_weight(J)]
          sage: [(x.weight(), x.energy_function()) for x in hw]
          [(-2*Lambda[0] + Lambda[2], 0), (-2*Lambda[0] + Lambda[1], 1), (0, 2)]
          sage: G = LS.digraph(index_set=J)
          sage: C = G.connected_components()
          sage: [all(c[0].energy_function()==a.energy_function() for a in c) for c in C]
          [True, True, True]

          sage: R = RootSystem(CartanType(['G',2,1]).dual())
          sage: La = R.weight_space().basis()
          sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[2])
          sage: G = LS.digraph(index_set=[1,2])
          sage: C = G.connected_components()
          sage: [all(c[0].energy_function()==a.energy_function() for a in c) for c in C] # long time
          [True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True]

          sage: ct = CartanType(['BC',2,2]).dual()
          sage: R = RootSystem(ct)
          sage: La = R.weight_space().basis()
          sage: LS = crystals.ProjectedLevelZeroLSPaths(2*La[1]+La[2])
          sage: G = LS.digraph(index_set=R.cartan_type().classical().index_set())
          sage: C = G.connected_components()
          sage: [all(c[0].energy_function()==a.energy_function() for a in c) for c in C] # long time
          [True, True, True, True, True, True, True, True, True, True, True]

          sage: R = RootSystem(['BC',2,2])
          sage: La = R.weight_space().basis()
          sage: LS = crystals.ProjectedLevelZeroLSPaths(2*La[1]+La[2])
          sage: G = LS.digraph(index_set=R.cartan_type().classical().index_set())
          sage: C = G.connected_components()
          sage: [all(c[0].energy_function()==a.energy_function() for a in c) for c in C] # long time
          [True, True, True, True, True, True, True, True, True, True, True, True, True, True, True,
          True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True]

    `scalar_factors`()

      Obtain the scalar factors for `self`.

      Each LS path (or `self`) can be written as a piecewise linear map

      \[\pi(t) = \sum_{u'=1}^{u-1} (\sigma_{u'} - \sigma_{u'-1}) \nu_{u'} + (t-\sigma_{u-1}) \nu_{u}\]

      for \(0<\sigma_1<\sigma_2<\cdots<\sigma_s=1\) and \(\sigma_{u-1} \le t \le \sigma_{u}\) and \(1 \le u \le s\). This method returns the tuple of \((\sigma_1,\ldots,\sigma_s)\).

      EXAMPLES:

          sage: R = RootSystem(['C',3,1])
          sage: La = R.weight_space().basis()
          sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[3])
          sage: b = LS.module_generators[0]
          sage: b.scalar_factors()
          [1]
          sage: c = b.f(1).f(3).f(2)
          sage: c.scalar_factors()
          [1/3, 1]

    `weyl_group_representation`()

      Transforms the weights in the LS path `self` to elements in the Weyl group.

      Each LS path can be written as the piecewise linear map:

      \[\pi(t) = \sum_{u'=1}^{u-1} (\sigma_{u'} - \sigma_{u'-1}) \nu_{u'} + (t-\sigma_{u-1}) \nu_{u}\]

      for \(0<\sigma_1<\sigma_2<\cdots<\sigma_s=1\) and \(\sigma_{u-1} \le t \le \sigma_{u}\) and \(1 \le u \le s\). Each weight \(\nu_u\) is also associated to a Weyl group element. This method returns the list of Weyl group elements associated to the \(\nu_u\) for \(1\le u\le s\).

      EXAMPLES:

          sage: R = RootSystem(['C',3,1])
          sage: La = R.weight_space().basis()
          sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[3])
          sage: b = LS.module_generators[0]
          sage: c = b.f(1).f(3).f(2)
          sage: c.weyl_group_representation()
          [s2*s3*s1, s3*s1]

  `CrystalOfProjectedLevelZeroLSPaths.``is_perfect`(level=1)

    Checks whether the crystal `self` is perfect (of level `level`).

    INPUT:

    - `level` – (default: 1) positive integer

    A crystal \(\mathcal{B}\) is perfect of level \(\ell\) if:

    1. \(\mathcal{B}\) is isomorphic to the crystal graph of a finite-dimensional \(U_q^{'}(\mathfrak{g})\)-module.
    2. \(\mathcal{B}\otimes \mathcal{B}\) is connected.
    3. There exists a \(\lambda\in X\), such that \(\mathrm{wt}(\mathcal{B}) \subset \lambda + \sum_{i\in I} \mathbb{Z}_{\le 0} \alpha_i\) and there is a unique element in \(\mathcal{B}\) of classical weight \(\lambda\).
    4. \(\forall b \in \mathcal{B}, \mathrm{level}(\varepsilon (b)) \geq \ell\).
    5. \(\forall \Lambda\) dominant weights of level \(\ell\), there exist unique elements \(b_{\Lambda}, b^{\Lambda} \in \mathcal{B}\), such that \(\varepsilon ( b_{\Lambda}) = \Lambda = \varphi( b^{\Lambda})\).

    Points (1)-(3) are known to hold. This method checks points (4) and (5).

    EXAMPLES:

        sage: C = CartanType(['C',2,1])
        sage: R = RootSystem(C)
        sage: La = R.weight_space().basis()
        sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1])
        sage: LS.is_perfect()
        False
        sage: LS = crystals.ProjectedLevelZeroLSPaths(La[2])
        sage: LS.is_perfect()
        True

        sage: C = CartanType(['E',6,1])
        sage: R = RootSystem(C)
        sage: La = R.weight_space().basis()
        sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1])
        sage: LS.is_perfect()
        True
        sage: LS.is_perfect(2)
        False

        sage: C = CartanType(['D',4,1])
        sage: R = RootSystem(C)
        sage: La = R.weight_space().basis()
        sage: all(crystals.ProjectedLevelZeroLSPaths(La[i]).is_perfect() for i in [1,2,3,4])
        True

        sage: C = CartanType(['A',6,2])
        sage: R = RootSystem(C)
        sage: La = R.weight_space().basis()
        sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[2])
        sage: LS.is_perfect()
        True
        sage: LS.is_perfect(2)
        False

  `CrystalOfProjectedLevelZeroLSPaths.``one_dimensional_configuration_sum`(q=None, group_components=True)

    Compute the one-dimensional configuration sum.

    INPUT:

    - `q` – (default: `None`) a variable or `None`; if `None`, a variable `q` is set in the code
    - `group_components` – (default: `True`) boolean; if `True`, then the terms are grouped by classical component

    The one-dimensional configuration sum is the sum of the weights of all elements in the crystal weighted by the energy function. For untwisted types it uses the parabolic quantum Bruhat graph, see [LNSSS2013]. In the dual-of-untwisted case, the parabolic quantum Bruhat graph is defined by exchanging the roles of roots and coroots (which is still conjectural at this point).

    EXAMPLES:

        sage: R = RootSystem(['A',2,1])
        sage: La = R.weight_space().basis()
        sage: LS = crystals.ProjectedLevelZeroLSPaths(2*La[1])
        sage: LS.one_dimensional_configuration_sum() # long time
        B[-2*Lambda[1] + 2*Lambda[2]] + (q+1)*B[-Lambda[1]]
        + (q+1)*B[Lambda[1] - Lambda[2]] + B[2*Lambda[1]] + B[-2*Lambda[2]] + (q+1)*B[Lambda[2]]
        sage: R.<t> = ZZ[]
        sage: LS.one_dimensional_configuration_sum(t, False) # long time
        B[-2*Lambda[1] + 2*Lambda[2]] + (t+1)*B[-Lambda[1]] + (t+1)*B[Lambda[1] - Lambda[2]]
        + B[2*Lambda[1]] + B[-2*Lambda[2]] + (t+1)*B[Lambda[2]]

    TESTS:

        sage: R = RootSystem(['B',3,1])
        sage: La = R.weight_space().basis()
        sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[2])
        sage: LS.one_dimensional_configuration_sum() == LS.one_dimensional_configuration_sum(group_components=False) # long time
        True
        sage: K1 = crystals.KirillovReshetikhin(['B',3,1],1,1)
        sage: K2 = crystals.KirillovReshetikhin(['B',3,1],2,1)
        sage: T = crystals.TensorProduct(K2,K1)
        sage: T.one_dimensional_configuration_sum() == LS.one_dimensional_configuration_sum() # long time
        True

        sage: R = RootSystem(['D',4,2])
        sage: La = R.weight_space().basis()
        sage: LS = crystals.ProjectedLevelZeroLSPaths(La[1]+La[2])
        sage: K1 = crystals.KirillovReshetikhin(['D',4,2],1,1)
        sage: K2 = crystals.KirillovReshetikhin(['D',4,2],2,1)
        sage: T = crystals.TensorProduct(K2,K1)
        sage: T.one_dimensional_configuration_sum() == LS.one_dimensional_configuration_sum() # long time
        True

        sage: R = RootSystem(['A',5,2])
        sage: La = R.weight_space().basis()
        sage: LS = crystals.ProjectedLevelZeroLSPaths(3*La[1])
        sage: K1 = crystals.KirillovReshetikhin(['A',5,2],1,1)
        sage: T = crystals.TensorProduct(K1,K1,K1)
        sage: T.one_dimensional_configuration_sum() == LS.one_dimensional_configuration_sum() # long time
        True

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