Merge branch 'public/15187' into 7.1.b5

src/sage/combinat/kac_polynomial.py

@@ -0,0 +1,138 @@
+r"""
+Kac Polynomials and DT-Invariants
+"""
+#*****************************************************************************
+#       Copyright (C) 2013 Travis Scrimshaw <tscrim at ucdavis.edu>,
+#
+#  Distributed under the terms of the GNU General Public License (GPL)
+#
+#    This code is distributed in the hope that it will be useful,
+#    but WITHOUT ANY WARRANTY; without even the implied warranty of
+#    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+#    General Public License for more details.
+#
+#  The full text of the GPL is available at:
+#
+#                  http://www.gnu.org/licenses/
+#*****************************************************************************
+
+from sage.misc.all import prod
+from sage.rings.all import ZZ
+from sage.combinat.sf.sf import SymmetricFunctions
+from sage.combinat.partition_tuple import PartitionTuple
+from sage.combinat.partition import Partitions
+from sage.rings.arith import moebius
+from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
+import itertools
+
+
+def cohomology_polynomial(Q, mu, var=None):
+    r"""
+    Return the cohomology polynomial of the quiver `Q` .
+
+    INPUT:
+
+    - `Q` -- a quiver as a digraph
+    - ``mu`` -- a partition tuple
+    - `var` -- (optional) the variable `q`
+
+    The cohomology polynomial is given by:
+
+    .. MATH::
+
+        \sum_i \dim\left( H_c^{2i}(\mathcal{Q}_{\tilde{v}} ; \CC)_{\epsilon
+        \chi^{\mu}} \right) q^{i - d_{\tilde{v}}} = \mathbb{H}_{\mu}^s(q)
+
+    where `\widetilde{v}` is the extended dimension vector given
+    in [HLRV2012]_, `\epsilon` is the sign character, and
+
+    .. MATH::
+
+        \mathbb{H}_{\mu}^s(q) := \bigl\langle \mathbb{H}(x_1, \ldots, x_r; q),
+        s_{\mu} \bigr\rangle,
+
+    where `\mathbb{H}` is given by Equation (1.4) in [HLRV2012]_.
+
+    EXAMPLES::
+
+        sage: from sage.combinat.kac_polynomial import cohomology_polynomial
+        sage: cohomology_polynomial(DiGraph([[1],[]]), PartitionTuple([[1]]))
+        1?
+
+    REFERENCES:
+
+    .. [HLRV2012] Tamas Hausel, Emmanuel Letellier, and Fernando
+       Rodriguez-Villegas. *Positivity of Kac polynomials and DT-invariants
+       for quivers*. (2012). :arxiv:`1204.2375v1`.
+    """
+    R = PolynomialRing(ZZ, 'q')
+    q = R.gen()
+    HLP = SymmetricFunctions(R).hall_littlewood(q).P()
+    n = ZZ(mu.size())
+    r = mu.level()
+    fac_r = (-1) ** (r - 1) / r
+
+    ret = R.zero()
+    for d in n.divisors():  # We must have d dividing n
+        qd = q ** d
+        terms = R.zero()
+        # We only have a chance if each partition's size times d is equal to
+        # the size of the corresponding partition in mu
+        for pt in itertools.product(*[Partitions(p.size() // d)
+                                      for p in mu.components()]):
+            Z = zip(mu.components(), pt)
+            t = prod(HLP[p].frobenius(d).coefficient(m) for m, p in Z)
+            if t != 0:  # If there's something to do
+                t *= prod(qd ** pt[i].pairing(pt[j])
+                          for i, j in Q.edges(False))
+                t /= prod(qd ** la.pairing(la) *
+                          prod(1 - qd ** -j for mk in la.to_exp_dict().values()
+                               for j in range(1, mk + 1))
+                          for la in pt)
+                terms += t
+        ret += moebius(d) * terms / d
+
+    ret *= fac_r * (q - 1)
+
+    if var is None:
+        return ret
+    else:
+        return ret(q=var)
+
+
+def kac_polynomial(Q, v, q=None):
+    """
+    Return the Kac polynomial for the quiver `Q` and the dimension vector `v`.
+
+    INPUT:
+
+    - `Q` -- a quiver as a digraph
+    - `v` -- the dimension vector
+    - `q` -- (optional) the variable `q`
+
+    EXAMPLES::
+
+        sage: from sage.combinat.kac_polynomial import kac_polynomial
+        sage: kac_polynomial(DiGraph([[1],[]]), (1,))
+        1?
+    """
+    return cohomology_polynomial(Q, PartitionTuple([[x] for x in v]), q)
+
+
+def DT_invariant(Q, v, q=None):
+    """
+    Return the DT-invariant for the quiver `Q` and the dimension vector `v`.
+
+    INPUT:
+
+    - `Q` -- a quiver as a digraph
+    - `v` -- the dimension vector
+    - `q` -- (optional) the variable `q`
+
+    EXAMPLES::
+
+        sage: from sage.combinat.kac_polynomial import DT_invariant
+        sage: DT_invariant(DiGraph([[1],[]]), (1,))
+        1?
+    """
+    return cohomology_polynomial(Q, PartitionTuple([[1] * x for x in v]), q)

src/sage/combinat/partition.py

@@ -4641,6 +4641,37 @@ def coloring(i):
                               immutable=True, multiedges=True)
         return self.dual_equivalence_graph(directed, coloring)
 
+    def pairing(self, mu):
+        r"""
+        Return the pairing of ``self`` and ``mu``.
+
+        The pairing of partitions `\lambda` and `\mu` is given by:
+
+        .. MATH::
+
+            \langle \lambda, \mu \rangle = \sum_{i,j} \min(i, j) m_i(\lambda)
+            m_j(\mu)
+
+        where `m_i(\lambda)` denotes the multiplicity of `i` in `\lambda`.
+
+        EXAMPLES::
+
+            sage: la = Partition([3,1,1])
+            sage: mu = Partition([2,1])
+            sage: la.pairing(mu)
+            7
+            sage: mu.pairing(la)
+            7
+            sage: la.pairing(la)
+            11
+            sage: mu.pairing(mu)
+            5
+        """
+        return sum(min(i, j) * mla * mmu
+                   for i, mla in self.to_exp_dict().items()
+                   for j, mmu in mu.to_exp_dict().items())
+
+
 ##############
 # Partitions #
 ##############