Hall Algebras¶
AUTHORS:
- Travis Scrimshaw (2013-10-17): Initial version
-
class
sage.algebras.hall_algebra.
HallAlgebra
(base_ring, q, prefix='H')¶ Bases:
sage.combinat.free_module.CombinatorialFreeModule
The (classical) Hall algebra.
The (classical) Hall algebra over a commutative ring
with a parameter
is defined to be the free
-module with basis
, where
runs over all integer partitions. The algebra structure is given by a product defined by
where
is a Hall polynomial (see
hall_polynomial()
). The unity of this algebra is.
The (classical) Hall algebra is also known as the Hall-Steinitz algebra.
We can define an
-algebra isomorphism
from the
-algebra of symmetric functions (see
SymmetricFunctions
) to the (classical) Hall algebra by sending the-th elementary symmetric function
to
for every positive integer
. This isomorphism used to transport the Hopf algebra structure from the
-algebra of symmetric functions to the Hall algebra, thus making the latter a connected graded Hopf algebra. If
is a partition, then the preimage of the basis element
under this isomorphism is
, where
denotes the
-th Hall-Littlewood
-function, and where
.
See section 2.3 in [Schiffmann], and sections II.2 and III.3 in [Macdonald1995] (where our
is called
).
EXAMPLES:
sage: R.<q> = ZZ[] sage: H = HallAlgebra(R, q) sage: H[2,1]*H[1,1] H[3, 2] + (q+1)*H[3, 1, 1] + (q^2+q)*H[2, 2, 1] + (q^4+q^3+q^2)*H[2, 1, 1, 1] sage: H[2]*H[2,1] H[4, 1] + q*H[3, 2] + (q^2-1)*H[3, 1, 1] + (q^3+q^2)*H[2, 2, 1] sage: H[3]*H[1,1] H[4, 1] + q^2*H[3, 1, 1] sage: H[3]*H[2,1] H[5, 1] + q*H[4, 2] + (q^2-1)*H[4, 1, 1] + q^3*H[3, 2, 1]
We can rewrite the Hall algebra in terms of monomials of the elements
:
sage: I = H.monomial_basis() sage: H(I[2,1,1]) H[3, 1] + (q+1)*H[2, 2] + (2*q^2+2*q+1)*H[2, 1, 1] + (q^5+2*q^4+3*q^3+3*q^2+2*q+1)*H[1, 1, 1, 1] sage: I(H[2,1,1]) I[3, 1] + (-q^3-q^2-q-1)*I[4]
The isomorphism between the Hall algebra and the symmetric functions described above is implemented as a coercion:
sage: R = PolynomialRing(ZZ, 'q').fraction_field() sage: q = R.gen() sage: H = HallAlgebra(R, q) sage: e = SymmetricFunctions(R).e() sage: e(H[1,1,1]) 1/q^3*e[3]
We can also do computations with any special value of
q
, such asor
or (most commonly) a prime power. Here is an example using a prime:
sage: H = HallAlgebra(ZZ, 2) sage: H[2,1]*H[1,1] H[3, 2] + 3*H[3, 1, 1] + 6*H[2, 2, 1] + 28*H[2, 1, 1, 1] sage: H[3,1]*H[2] H[5, 1] + H[4, 2] + 6*H[3, 3] + 3*H[4, 1, 1] + 8*H[3, 2, 1] sage: H[2,1,1]*H[3,1] H[5, 2, 1] + 2*H[4, 3, 1] + 6*H[4, 2, 2] + 7*H[5, 1, 1, 1] + 19*H[4, 2, 1, 1] + 24*H[3, 3, 1, 1] + 48*H[3, 2, 2, 1] + 105*H[4, 1, 1, 1, 1] + 224*H[3, 2, 1, 1, 1] sage: I = H.monomial_basis() sage: H(I[2,1,1]) H[3, 1] + 3*H[2, 2] + 13*H[2, 1, 1] + 105*H[1, 1, 1, 1] sage: I(H[2,1,1]) I[3, 1] - 15*I[4]
If
is set to
, the coercion to the symmetric functions sends
to
:
sage: H = HallAlgebra(QQ, 1) sage: H[2,1] * H[2,1] H[4, 2] + 2*H[3, 3] + 2*H[4, 1, 1] + 2*H[3, 2, 1] + 6*H[2, 2, 2] + 4*H[2, 2, 1, 1] sage: m = SymmetricFunctions(QQ).m() sage: m[2,1] * m[2,1] 4*m[2, 2, 1, 1] + 6*m[2, 2, 2] + 2*m[3, 2, 1] + 2*m[3, 3] + 2*m[4, 1, 1] + m[4, 2] sage: m(H[3,1]) m[3, 1]
We can set
to
(but should keep in mind that we don’t get the Schur functions this way):
sage: H = HallAlgebra(QQ, 0) sage: H[2,1] * H[2,1] H[4, 2] + H[3, 3] + H[4, 1, 1] - H[3, 2, 1] - H[3, 1, 1, 1]
TESTS:
The coefficients are actually Laurent polynomials in general, so we don’t have to work over the fraction field of
. This didn’t work before trac ticket #15345:
sage: R.<q> = LaurentPolynomialRing(ZZ) sage: H = HallAlgebra(R, q) sage: I = H.monomial_basis() sage: hi = H(I[2,1]); hi H[2, 1] + (1+q+q^2)*H[1, 1, 1] sage: hi.parent() is H True sage: h22 = H[2]*H[2]; h22 H[4] - (1-q)*H[3, 1] + (q+q^2)*H[2, 2] sage: h22.parent() is H True sage: e = SymmetricFunctions(R).e() sage: e(H[1,1,1]) (q^-3)*e[3]
REFERENCES:
[Schiffmann] (1, 2) Oliver Schiffmann. Lectures on Hall algebras. Arxiv 0611617v2. -
class
Element
(M, x)¶ Bases:
sage.combinat.free_module.CombinatorialFreeModuleElement
Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s
__call__()
method.TESTS:
sage: F = CombinatorialFreeModule(QQ, ['a','b','c']) sage: B = F.basis() sage: f = B['a'] + 3*B['c']; f B['a'] + 3*B['c'] sage: f == loads(dumps(f)) True
-
scalar
(y)¶ Return the scalar product of
self
andy
.The scalar product is given by
where
is given by
where
and
.
Note that
can be interpreted as the number of automorphisms of a certain object in a category corresponding to
. See Lemma 2.8 in [Schiffmann] for details.
EXAMPLES:
sage: R.<q> = ZZ[] sage: H = HallAlgebra(R, q) sage: H[1].scalar(H[1]) 1/(q - 1) sage: H[2].scalar(H[2]) 1/(q^2 - q) sage: H[2,1].scalar(H[2,1]) 1/(q^5 - 2*q^4 + q^3) sage: H[1,1,1,1].scalar(H[1,1,1,1]) 1/(q^16 - q^15 - q^14 + 2*q^11 - q^8 - q^7 + q^6) sage: H.an_element().scalar(H.an_element()) (4*q^2 + 9)/(q^2 - q)
-
-
HallAlgebra.
antipode_on_basis
(la)¶ Return the antipode of the basis element indexed by
la
.EXAMPLES:
sage: R = PolynomialRing(ZZ, 'q').fraction_field() sage: q = R.gen() sage: H = HallAlgebra(R, q) sage: H.antipode_on_basis(Partition([1,1])) 1/q*H[2] + 1/q*H[1, 1] sage: H.antipode_on_basis(Partition([2])) -1/q*H[2] + ((q^2-1)/q)*H[1, 1] sage: R.<q> = LaurentPolynomialRing(ZZ) sage: H = HallAlgebra(R, q) sage: H.antipode_on_basis(Partition([1,1])) (q^-1)*H[2] + (q^-1)*H[1, 1] sage: H.antipode_on_basis(Partition([2])) -(q^-1)*H[2] - (q^-1-q)*H[1, 1]
-
HallAlgebra.
coproduct_on_basis
(la)¶ Return the coproduct of the basis element indexed by
la
.EXAMPLES:
sage: R = PolynomialRing(ZZ, 'q').fraction_field() sage: q = R.gen() sage: H = HallAlgebra(R, q) sage: H.coproduct_on_basis(Partition([1,1])) H[] # H[1, 1] + 1/q*H[1] # H[1] + H[1, 1] # H[] sage: H.coproduct_on_basis(Partition([2])) H[] # H[2] + ((q-1)/q)*H[1] # H[1] + H[2] # H[] sage: H.coproduct_on_basis(Partition([2,1])) H[] # H[2, 1] + ((q^2-1)/q^2)*H[1] # H[1, 1] + 1/q*H[1] # H[2] + ((q^2-1)/q^2)*H[1, 1] # H[1] + 1/q*H[2] # H[1] + H[2, 1] # H[] sage: R.<q> = LaurentPolynomialRing(ZZ) sage: H = HallAlgebra(R, q) sage: H.coproduct_on_basis(Partition([2])) H[] # H[2] - (q^-1-1)*H[1] # H[1] + H[2] # H[] sage: H.coproduct_on_basis(Partition([2,1])) H[] # H[2, 1] - (q^-2-1)*H[1] # H[1, 1] + (q^-1)*H[1] # H[2] - (q^-2-1)*H[1, 1] # H[1] + (q^-1)*H[2] # H[1] + H[2, 1] # H[]
-
HallAlgebra.
counit
(x)¶ Return the counit of the element
x
.EXAMPLES:
sage: R = PolynomialRing(ZZ, 'q').fraction_field() sage: q = R.gen() sage: H = HallAlgebra(R, q) sage: H.counit(H.an_element()) 2
-
HallAlgebra.
monomial_basis
()¶ Return the basis of the Hall algebra given by monomials in the
.
EXAMPLES:
sage: R.<q> = ZZ[] sage: H = HallAlgebra(R, q) sage: H.monomial_basis() Hall algebra with q=q over Univariate Polynomial Ring in q over Integer Ring in the monomial basis
-
HallAlgebra.
one_basis
()¶ Return the index of the basis element
.
EXAMPLES:
sage: R.<q> = ZZ[] sage: H = HallAlgebra(R, q) sage: H.one_basis() []
-
HallAlgebra.
product_on_basis
(mu, la)¶ Return the product of the two basis elements indexed by
mu
andla
.EXAMPLES:
sage: R.<q> = ZZ[] sage: H = HallAlgebra(R, q) sage: H.product_on_basis(Partition([1,1]), Partition([1])) H[2, 1] + (q^2+q+1)*H[1, 1, 1] sage: H.product_on_basis(Partition([2,1]), Partition([1,1])) H[3, 2] + (q+1)*H[3, 1, 1] + (q^2+q)*H[2, 2, 1] + (q^4+q^3+q^2)*H[2, 1, 1, 1] sage: H.product_on_basis(Partition([3,2]), Partition([2,1])) H[5, 3] + (q+1)*H[4, 4] + q*H[5, 2, 1] + (2*q^2-1)*H[4, 3, 1] + (q^3+q^2)*H[4, 2, 2] + (q^4+q^3)*H[3, 3, 2] + (q^4-q^2)*H[4, 2, 1, 1] + (q^5+q^4-q^3-q^2)*H[3, 3, 1, 1] + (q^6+q^5)*H[3, 2, 2, 1] sage: H.product_on_basis(Partition([3,1,1]), Partition([2,1])) H[5, 2, 1] + q*H[4, 3, 1] + (q^2-1)*H[4, 2, 2] + (q^3+q^2)*H[3, 3, 2] + (q^2+q+1)*H[5, 1, 1, 1] + (2*q^3+q^2-q-1)*H[4, 2, 1, 1] + (q^4+2*q^3+q^2)*H[3, 3, 1, 1] + (q^5+q^4)*H[3, 2, 2, 1] + (q^6+q^5+q^4-q^2-q-1)*H[4, 1, 1, 1, 1] + (q^7+q^6+q^5)*H[3, 2, 1, 1, 1]
-
class
-
class
sage.algebras.hall_algebra.
HallAlgebraMonomials
(base_ring, q, prefix='I')¶ Bases:
sage.combinat.free_module.CombinatorialFreeModule
The classical Hall algebra given in terms of monomials in the
.
We first associate a monomial
with the composition
. However since
commutes with
, the basis is indexed by partitions.
EXAMPLES:
We use the fraction field of
for our initial example:
sage: R = PolynomialRing(ZZ, 'q').fraction_field() sage: q = R.gen() sage: H = HallAlgebra(R, q) sage: I = H.monomial_basis()
We check that the basis conversions are mutually inverse:
sage: all(H(I(H[p])) == H[p] for i in range(7) for p in Partitions(i)) True sage: all(I(H(I[p])) == I[p] for i in range(7) for p in Partitions(i)) True
Since Laurent polynomials are sufficient, we run the same check with the Laurent polynomial ring
:
sage: R.<q> = LaurentPolynomialRing(ZZ) sage: H = HallAlgebra(R, q) sage: I = H.monomial_basis() sage: all(H(I(H[p])) == H[p] for i in range(6) for p in Partitions(i)) # long time True sage: all(I(H(I[p])) == I[p] for i in range(6) for p in Partitions(i)) # long time True
We can also convert to the symmetric functions. The natural basis corresponds to the Hall-Littlewood basis (up to a renormalization and an inversion of the
parameter), and this basis corresponds to the elementary basis (up to a renormalization):
sage: Sym = SymmetricFunctions(R) sage: e = Sym.e() sage: e(I[2,1]) (q^-1)*e[2, 1] sage: e(I[4,2,2,1]) (q^-8)*e[4, 2, 2, 1] sage: HLP = Sym.hall_littlewood(q).P() sage: H(I[2,1]) H[2, 1] + (1+q+q^2)*H[1, 1, 1] sage: HLP(e[2,1]) (1+q+q^2)*HLP[1, 1, 1] + HLP[2, 1] sage: all( e(H[lam]) == q**-sum([i * x for i, x in enumerate(lam)]) ....: * e(HLP[lam]).map_coefficients(lambda p: p(q**(-1))) ....: for lam in Partitions(4) ) True
We can also do computations using a prime power:
sage: H = HallAlgebra(ZZ, 3) sage: I = H.monomial_basis() sage: i_elt = I[2,1]*I[1,1]; i_elt I[2, 1, 1, 1] sage: H(i_elt) H[4, 1] + 7*H[3, 2] + 37*H[3, 1, 1] + 136*H[2, 2, 1] + 1495*H[2, 1, 1, 1] + 62920*H[1, 1, 1, 1, 1]
-
class
Element
(M, x)¶ Bases:
sage.combinat.free_module.CombinatorialFreeModuleElement
Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s
__call__()
method.TESTS:
sage: F = CombinatorialFreeModule(QQ, ['a','b','c']) sage: B = F.basis() sage: f = B['a'] + 3*B['c']; f B['a'] + 3*B['c'] sage: f == loads(dumps(f)) True
-
scalar
(y)¶ Return the scalar product of
self
andy
.The scalar product is computed by converting into the natural basis.
EXAMPLES:
sage: R.<q> = ZZ[] sage: I = HallAlgebra(R, q).monomial_basis() sage: I[1].scalar(I[1]) 1/(q - 1) sage: I[2].scalar(I[2]) 1/(q^4 - q^3 - q^2 + q) sage: I[2,1].scalar(I[2,1]) (2*q + 1)/(q^6 - 2*q^5 + 2*q^3 - q^2) sage: I[1,1,1,1].scalar(I[1,1,1,1]) 24/(q^4 - 4*q^3 + 6*q^2 - 4*q + 1) sage: I.an_element().scalar(I.an_element()) (4*q^4 - 4*q^2 + 9)/(q^4 - q^3 - q^2 + q)
-
-
HallAlgebraMonomials.
antipode_on_basis
(a)¶ Return the antipode of the basis element indexed by
a
.EXAMPLES:
sage: R = PolynomialRing(ZZ, 'q').fraction_field() sage: q = R.gen() sage: I = HallAlgebra(R, q).monomial_basis() sage: I.antipode_on_basis(Partition([1])) -I[1] sage: I.antipode_on_basis(Partition([2])) 1/q*I[1, 1] - I[2] sage: I.antipode_on_basis(Partition([2,1])) -1/q*I[1, 1, 1] + I[2, 1] sage: R.<q> = LaurentPolynomialRing(ZZ) sage: I = HallAlgebra(R, q).monomial_basis() sage: I.antipode_on_basis(Partition([2,1])) -(q^-1)*I[1, 1, 1] + I[2, 1]
-
HallAlgebraMonomials.
coproduct_on_basis
(a)¶ Return the coproduct of the basis element indexed by
a
.EXAMPLES:
sage: R = PolynomialRing(ZZ, 'q').fraction_field() sage: q = R.gen() sage: I = HallAlgebra(R, q).monomial_basis() sage: I.coproduct_on_basis(Partition([1])) I[] # I[1] + I[1] # I[] sage: I.coproduct_on_basis(Partition([2])) I[] # I[2] + 1/q*I[1] # I[1] + I[2] # I[] sage: I.coproduct_on_basis(Partition([2,1])) I[] # I[2, 1] + 1/q*I[1] # I[1, 1] + I[1] # I[2] + 1/q*I[1, 1] # I[1] + I[2] # I[1] + I[2, 1] # I[] sage: R.<q> = LaurentPolynomialRing(ZZ) sage: I = HallAlgebra(R, q).monomial_basis() sage: I.coproduct_on_basis(Partition([2,1])) I[] # I[2, 1] + (q^-1)*I[1] # I[1, 1] + I[1] # I[2] + (q^-1)*I[1, 1] # I[1] + I[2] # I[1] + I[2, 1] # I[]
-
HallAlgebraMonomials.
counit
(x)¶ Return the counit of the element
x
.EXAMPLES:
sage: R = PolynomialRing(ZZ, 'q').fraction_field() sage: q = R.gen() sage: I = HallAlgebra(R, q).monomial_basis() sage: I.counit(I.an_element()) 2
-
HallAlgebraMonomials.
one_basis
()¶ Return the index of the basis element
.
EXAMPLES:
sage: R.<q> = ZZ[] sage: I = HallAlgebra(R, q).monomial_basis() sage: I.one_basis() []
-
HallAlgebraMonomials.
product_on_basis
(a, b)¶ Return the product of the two basis elements indexed by
a
andb
.EXAMPLES:
sage: R.<q> = ZZ[] sage: I = HallAlgebra(R, q).monomial_basis() sage: I.product_on_basis(Partition([4,2,1]), Partition([3,2,1])) I[4, 3, 2, 2, 1, 1]
-
class
-
sage.algebras.hall_algebra.
transpose_cmp
(x, y)¶ Compare partitions
x
andy
in transpose dominance order.We say partitions
and
satisfy
in transpose dominance order if for all
we have:
where
denotes the number of appearances of
in
, and
denotes the number of appearances of
in
.
Equivalently,
if the conjugate of the partition
dominates the conjugate of the partition
.
Since this is a partial ordering, we fallback to lex ordering
if we cannot compare in the transpose order.
EXAMPLES:
sage: from sage.algebras.hall_algebra import transpose_cmp sage: transpose_cmp(Partition([4,3,1]), Partition([3,2,2,1])) -1 sage: transpose_cmp(Partition([2,2,1]), Partition([3,2])) 1 sage: transpose_cmp(Partition([4,1,1]), Partition([4,1,1])) 0