Hall Algebras

AUTHORS:

  • Travis Scrimshaw (2013-10-17): Initial version
sage.algebras.hall_algebra.HallAlgebra

The (classical) Hall algebra.

The (classical) Hall algebra over a commutative ring \(R\) with a parameter \(q \in R\) is defined to be the free \(R\)-module with basis \((I_\lambda)\), where \(\lambda\) runs over all integer partitions. The algebra structure is given by a product defined by

\[I_\mu \cdot I_\lambda = \sum_\nu P^{\nu}_{\mu, \lambda}(q) I_\nu,\]

where \(P^{\nu}_{\mu, \lambda}\) is a Hall polynomial (see hall_polynomial()). The unity of this algebra is \(I_{\emptyset}\).

The (classical) Hall algebra is also known as the Hall-Steinitz algebra.

We can define an \(R\)-algebra isomorphism \(\Phi\) from the \(R\)-algebra of symmetric functions (see SymmetricFunctions) to the (classical) Hall algebra by sending the \(r\)-th elementary symmetric function \(e_r\) to \(q^{r(r-1)/2} I_{(1^r)}\) for every positive integer \(r\). This isomorphism used to transport the Hopf algebra structure from the \(R\)-algebra of symmetric functions to the Hall algebra, thus making the latter a connected graded Hopf algebra. If \(\lambda\) is a partition, then the preimage of the basis element \(I_{\lambda}\) under this isomorphism is \(q^{n(\lambda)} P_{\lambda}(x; q^{-1})\), where \(P_{\lambda}\) denotes the \(\lambda\)-th Hall-Littlewood \(P\)-function, and where \(n(\lambda) = \sum_i (i - 1) \lambda_i\).

See section 2.3 in [Sch2006], and sections II.2 and III.3 in [Macdonald1995] (where our \(I_{\lambda}\) is called \(u_{\lambda}\)).

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 \(I_{(1^r)}\):

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 as \(0\) or \(1\) 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 \(q\) is set to \(1\), the coercion to the symmetric functions sends \(I_{\lambda}\) to \(m_{\lambda}\):

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 \(q\) to \(0\) (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]
sage.algebras.hall_algebra.HallAlgebraMonomials

The classical Hall algebra given in terms of monomials in the \(I_{(1^r)}\).

We first associate a monomial \(I_{(1^{r_1})} I_{(1^{r_2})} \cdots I_{(1^{r_k})}\) with the composition \((r_1, r_2, \ldots, r_k)\). However since \(I_{(1^r)}\) commutes with \(I_{(1^s)}\), the basis is indexed by partitions.

EXAMPLES:

We use the fraction field of \(\ZZ[q]\) 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 \(\ZZ[q, q^{-1}]\):

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 \(q\) 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]
sage.algebras.hall_algebra.transpose_cmp(x, y)

Compare partitions x and y in transpose dominance order.

We say partitions \(\mu\) and \(\lambda\) satisfy \(\mu \prec \lambda\) in transpose dominance order if for all \(i \geq 1\) we have:

\[l_1 + 2 l_2 + \cdots + (i-1) l_{i-1} + i(l_i + l_{i+1} + \cdots) \leq m_1 + 2 m_2 + \cdots + (i-1) m_{i-1} + i(m_i + m_{i+1} + \cdots),\]

where \(l_k\) denotes the number of appearances of \(k\) in \(\lambda\), and \(m_k\) denotes the number of appearances of \(k\) in \(\mu\).

Equivalently, \(\mu \prec \lambda\) if the conjugate of the partition \(\mu\) dominates the conjugate of the partition \(\lambda\).

Since this is a partial ordering, we fallback to lex ordering \(\mu <_L \lambda\) 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