Hall Polynomials

sage.combinat.hall_polynomial.hall_polynomial(nu, mu, la, q=None)

Return the (classical) Hall polynomial \(P^{\nu}_{\mu,\lambda}\) (where \(\nu\), \(\mu\) and \(\lambda\) are the inputs nu, mu and la).

Let \(\nu,\mu,\lambda\) be partitions. The Hall polynomial \(P^{\nu}_{\mu,\lambda}(q)\) (in the indeterminate \(q\)) is defined as follows: Specialize \(q\) to a prime power, and consider the category of \(\GF{q}\)-vector spaces with a distinguished nilpotent endomorphism. The morphisms in this category shall be the linear maps commuting with the distinguished endomorphisms. The type of an object in the category will be the Jordan type of the distinguished endomorphism; this is a partition. Now, if \(N\) is any fixed object of type \(\nu\) in this category, then the polynomial \(P^{\nu}_{\mu,\lambda}(q)\) specialized at the prime power \(q\) counts the number of subobjects \(L\) of \(N\) having type \(\lambda\) such that the quotient object \(N / L\) has type \(\mu\). This determines the values of the polynomial \(P^{\nu}_{\mu,\lambda}\) at infinitely many points (namely, at all prime powers), and hence \(P^{\nu}_{\mu,\lambda}\) is uniquely determined. The degree of this polynomial is at most \(n(\nu) - n(\lambda) - n(\mu)\), where \(n(\kappa) = \sum_i (i-1) \kappa_i\) for every partition \(\kappa\). (If this is negative, then the polynomial is zero.)

These are the structure coefficients of the (classical) Hall algebra.

If \(\lvert \nu \rvert \neq \lvert \mu \rvert + \lvert \lambda \rvert\), then we have \(P^{\nu}_{\mu,\lambda} = 0\). More generally, if the Littlewood-Richardson coefficient \(c^{\nu}_{\mu,\lambda}\) vanishes, then \(P^{\nu}_{\mu,\lambda} = 0\).

The Hall polynomials satisfy the symmetry property \(P^{\nu}_{\mu,\lambda} = P^{\nu}_{\lambda,\mu}\).


If \(\lambda = (1^r)\) and \(\lvert \nu \rvert = \lvert \mu \rvert + \lvert \lambda \rvert\), then we compute \(P^{\nu}_{\mu,\lambda}\) as follows (cf. Example 2.4 in [Sch2006]):

First, write \(\nu = (1^{l_1}, 2^{l_2}, \ldots, n^{l_n})\), and define a sequence \(r = r_0 \geq r_1 \geq \cdots \geq r_n\) such that

\[\mu = \left( 1^{l_1 - r_0 + 2r_1 - r_2}, 2^{l_2 - r_1 + 2r_2 - r_3}, \ldots , (n-1)^{l_{n-1} - r_{n-2} + 2r_{n-1} - r_n}, n^{l_n - r_{n-1} + r_n} \right).\]

Thus if \(\mu = (1^{m_1}, \ldots, n^{m_n})\), we have the following system of equations:

\[\begin{split}\begin{aligned} m_1 & = l_1 - r + 2r_1 - r_2, \\ m_2 & = l_2 - r_1 + 2r_2 - r_3, \\ & \vdots , \\ m_{n-1} & = l_{n-1} - r_{n-2} + 2r_{n-1} - r_n, \\ m_n & = l_n - r_{n-1} + r_n \end{aligned}\end{split}\]

and solving for \(r_i\) and back substituting we obtain the equations:

\[\begin{split}\begin{aligned} r_n & = r_{n-1} + m_n - l_n, \\ r_{n-1} & = r_{n-2} + m_{n-1} - l_{n-1} + m_n - l_n, \\ & \vdots , \\ r_1 & = r + \sum_{k=1}^n (m_k - l_k), \end{aligned}\end{split}\]

or in general we have the recursive equation:

\[r_i = r_{i-1} + \sum_{k=i}^n (m_k - l_k).\]

This, combined with the condition that \(r_0 = r\), determines the \(r_i\) uniquely (recursively). Next we define

\[t = (r_{n-2} - r_{n-1})(l_n - r_{n-1}) + (r_{n-3} - r_{n-2})(l_{n-1} + l_n - r_{n-2}) + \cdots + (r_0 - r_1)(l_2 + \cdots + l_n - r_1),\]

and with these notations we have

\[P^{\nu}_{\mu,(1^r)} = q^t \binom{l_n}{r_{n-1}}_q \binom{l_{n-1}}{r_{n-2} - r_{n-1}}_q \cdots \binom{l_1}{r_0 - r_1}_q.\]

To compute \(P^{\nu}_{\mu,\lambda}\) in general, we compute the product \(I_{\mu} I_{\lambda}\) in the Hall algebra and return the coefficient of \(I_{\nu}\).


sage: from sage.combinat.hall_polynomial import hall_polynomial
sage: hall_polynomial([1,1],[1],[1])
q + 1
sage: hall_polynomial([2],[1],[1])
sage: hall_polynomial([2,1],[2],[1])
sage: hall_polynomial([2,2,1],[2,1],[1,1])
q^2 + q
sage: hall_polynomial([2,2,2,1],[2,2,1],[1,1])
q^4 + q^3 + q^2
sage: hall_polynomial([3,2,2,1], [3,2], [2,1])
q^6 + q^5
sage: hall_polynomial([4,2,1,1], [3,1,1], [2,1])
2*q^3 + q^2 - q - 1
sage: hall_polynomial([4,2], [2,1], [2,1], 0)