J. ALGEBRAIC GEOMETRY 00 (XXXX) 000–000 S 1056-3911(XX)0000-0

THE DEGREE OF THE DISCRIMINANT OF IRREDUCIBLE REPRESENTATIONS

L. M. FEHÉR, A. NÉMETHI, AND R. RIMÁNYI

Abstract We present a formula for the degree of the discriminant of irreducible representations of a Lie group, in terms of the roots of the group and the highest weight of the representation. The proof uses equivariant cohomology techniques, namely, the theory of Thom polynomials, and a new method for their computation. We study the combinatorics of our formulas in various special cases.

1. Introduction

Let G be a complex connected reductive algebraic group, and let ρ : G → GL(V) be an irreducible algebraic representation. Then ρ induces an action of G on the projective space ℙ(V). This action has a single closed orbit, the orbit of the weight vector of the highest weight λ. E.g., for GL(n) acting on Λ^{k}ℂ^{n}, we get the Grassmannian Gr_{k}(ℂ^{n}). The dual ℙD_{λ} of this orbit (or the affine cone D_{λ} over it) is called the discriminant of ρ since it generalizes the classical discriminant. The goal of the present paper is to give a formula for the degree of the discriminant in terms of the highest weight of the representation ρ and the roots of G (Theorems 5.2 and 5.6).

The classical approach to find the degree of dual varieties is due to Kleiman [?] and Katz [?]. Their method, however, does not produce a formula in the general setting. Special cases were worked out by Holme [?], Lascoux [?], Boole, Tevelev [?], Gelfand-Kapranov-Zelevinsky [?, Ch.13,14], see a summary in [?, Ch.7]. De Concini and Weyman [?] showed that, if G is fixed, then for regular highest weights the formula for the degree of the discriminant is a polynomial with positive coefficients, and they calculated the constant term of this polynomial. A corollary of our result is an explicit form for this polynomial (Cor. 5.8) with the additional fact that the same polynomial calculates the corresponding degrees for non-regular highest weights as well (modulo an explicit factor).

Supported by NSF grant DMS-0304759, and OTKA 42769/46878 (2nd author), NSF grant DMS-0405723 (3rd author) and OTKA T046365MAT (1st and 3rd author)

Keywords: discriminant, degree, projective duality, Thom polynomials.

In special cases our formula can be expressed in terms of some basic concepts in the combinatorics of polynomials [?, ?], such as the Jacobi symmetrizer, divided difference operators, or the scalar product on the space of polynomials (Section 6).

For the group GL(n) we further simplify the formula in many special cases in Section 7.

The authors thank M. Kazarian, A. Knutson, S. Kumar, and A. Szenes for helpful discussions. The original proof of Theorem 4.2 was more complicated. We got the idea of a simpler proof from a lecture of A. Szenes. We thank the referees for valuable suggestions.

2. Degree and Thom polynomials

In this paper we will use cohomology with rational coefficients. The Lie groups we consider are complex connected reductive algebraic groups. All varieties are over the complex numbers ℂ and GL(n) = GL(n; ℂ) denotes the general linear group of ℂ^{n}.

2.1. Degree and cohomology. Suppose that Y is a smooth complex algebraic variety and X ⊂ Y is a closed subvariety of complex codimension d. Then X represents a cohomology class [X] in the cohomology group H^{2d}(Y). This class is called the Poincaré dual of X. The existence of this class and its basic properties are explained e.g. in [?]. If Y is the projective space ℙ^{n} then H^{∗}(Y) ≅ ℚ[x]/(x^{n+1}) and [X] = deg(X)x^{d}, where x is the class represented by a hyperplane. By definition, the cone CX ⊂ ℂ^{n+1} of X has the same degree.

2.2. Degree and equivariant cohomology. We would also like to express the degree in terms of equivariant cohomology. Let G be a complex connected reductive algebraic group (though some definitions and claims hold for more general groups as well). Let G act on a topological space Y. Then the equivariant cohomology ring H_{G}^{∗}(Y) is defined as the ordinary cohomology of the Borel construction B_{G}Y = EG ×_{G} Y. Here EG denotes the universal principal G-bundle over the classifying space BG. If Y is a smooth complex algebraic variety, G is a complex Lie group acting on Y and X ⊂ Y is a G-invariant subvariety of complex codimension d then X represents an equivariant cohomology class (sometimes called equivariant Poincaré dual or, if Y is contractible, Thom polynomial) [X] ∈ H_{G}^{2d}(Y). This class has following the universal property.

Let P → M be an algebraic principal G-bundle. Then P is classified by a map k : M → BG in other words we have the diagram

\xymatrix{P \ar[r]^{k̃} \ar[d] & EG \ar[d] \\ M \ar[r]^{k} & BG.}

We also have an associated bundle P ×_{G} Y → M and an induced map k̂ : P ×_{G} Y → B_{G}Y = EG ×_{G} Y. The universal property of [X] ∈ H_{G}^{2d}(Y) is that for every P → M the ordinary cohomology class [P ×_{G} X] represented by the subvariety P ×_{G} X of P ×_{G} Y we have

(2.1)  k̂^{∗}[X] = [P ×_{G} X].

It is easy to see that (2.1) characterizes the equivariant cohomology class [X] ∈ H^{∗}(BG ×_{G} Y) = H_{G}^{∗}(Y) uniquely. The existence of such a class is not so obvious, for details see [?] or [?].

Let Y = V be a complex vector space and let G = GL(1) act as scalars. If X ⊂ Y is a G-invariant subvariety of complex codimension d then X is the cone of a projective variety ℙX ⊂ ℙV. Then H_{G}^{∗}(V) ≅ ℚ[x] and [X] = deg(ℙX)x^{d}, where x is the class of a hyperplane. For more general complex Lie groups we can expect to extract the degree if G ‘contains the scalars’. As we will see, even this condition is not necessary: we can simply replace G with G × GL(1).

Suppose that ρ : G → GL(V) ‘contains the scalars’ i.e. there is a homomorphism h : GL(1) → G, such that ρ ∘ h is the scalar representation on V: for v ∈ V and z ∈ GL(1) we have ρ ∘ h(z)v = zv. By the basic properties of the equivariant Poincaré dual we have a ‘change of action’ formula:

(2.2)  [X]_{ρ∘h} = h^{∗}[X]_{ρ},

where

h^{∗} : H_{G}^{∗}(V) ≅ H^{∗}(BG) → H_{GL(1)}^{∗}(V) ≅ H^{∗}(BGL(1))

is induced by the map Bh : BGL(1) → BG classifying the principal G-bundle EGL(1)×_{h}G. The ρ and ρ∘h in the lower index indicates whether we take the G- or GL(1)-equivariant Poincaré dual of X. The ‘change of action’ formula implies that

deg(X)x^{d} = h^{∗}[X]_{ρ}.

It frequently happens that we can only find a homomorphism h : GL(1) → G, such that ρ ∘ h(z)v = z^{k}v for some non zero integer k. Then, by the same way we obtain

k^{d} deg(X)x^{d} = [X]_{ρ∘h} = h^{∗}[X]_{ρ}.

The calculation of h^{∗} is fairly simple. Suppose that m : GL(1)^{r} → G is a (parametrized) maximal complex torus of G. Then by Borel’s theorem [?, §27] H^{∗}(BG) is naturally isomorphic to the Weyl-invariant subring of H^{∗}(BT) (T is the image of m, a complex maximal torus of G), and H^{∗}(BT) can be identified with the symmetric algebra of the character group of T. Hence H^{∗}(BG) = ℚ[α_{1}, …, α_{r}]^{W}, where the α_{i}’s generate the weight lattice of G (one can identify α_{i} with the i^{th} projection π_{i} : GL(1)^{r} → GL(1)) and W is the Weyl group of G. Then [X] = p(α_{1}, …, α_{r}) for some homogeneous polynomial p of degree d. We can assume that the homomorphism h : GL(1) → G factors through m, i.e. h = m ∘ ϕ, where ϕ(z) = (z^{k_{1}}, …, z^{k_{r}}) for some integers k_{1}, …, k_{r}. Applying the ‘change of action’ formula (2.2) once more leads to

Proposition 2.1. For the polynomial p, and integers k, k_{1}, …, k_{r} defined above

deg(X) = p(k_{1}/k, …, k_{r}/k).

This innocent-looking statement provides a uniform approach for calculating the degree of degeneracy loci whenever a Chern–class formula is known. A similar but more involved argument (see [?]) calculates the equivariant Poincaré dual of ℙX in ℙV.

3. Calculation of the equivariant Poincaré dual

Now we reduce the problem of computing an equivariant Poincaré dual to computing an integral. We will need G-equivariant characteristic classes: the equivariant Chern classes c_{i}(E) ∈ H_{G}^{2i}(M) and the equivariant Euler class e(E) ∈ H_{G}^{2n}(M) of a G-equivariant complex vector bundle E → M of rank n are defined via the Borel construction. We have e(E) = c_{n}(E).

As in the case of ordinary cohomology, pushforward can be defined in the equivariant setting. An introduction can be found in [?]. Its properties are similar; for example if ϕ : X̃ → Y is a G-equivariant resolution of the dcodimensional invariant subvariety X ⊂ Y, then [X] = ϕ_{∗}1 ∈ H_{G}^{2d}(Y).

Let V be a vector space and X ⊂ V a subvariety. Suppose that M is a compact manifold and let X̃ → M be a sub-vector bundle S of M × V → M. Let π : M × V → V be the projection, i : X̃ ⊂ M × V the embedding, and let ϕ = π ∘ i, as in the diagram

\xymatrix{X̃ \ar[rrr]^{ϕ} \ar@{^{(}->}[rr]^{i} \ar[dr] &  & M × V \ar[dl] \ar[r]^{π} & V \\  & M &  & X. \ar@{^{(}->}[u]}

Proposition 3.1. Suppose that G acts on all spaces in the diagram above, and that all maps are G-invariant. Let Q be the quotient bundle V ⊖ S over M. We have

(3.1)  π_{∗}i_{∗}1 = ∫_{M} e(Q)  ∈ H_{G}^{∗}(V) = H^{∗}(BG).

If ϕ is a resolution of X ⊂ V then

(3.2)  [X] = ∫_{M} e(Q)  ∈ H_{G}^{∗}(V) = H^{∗}(BG).

Proof. Observe that i_{∗}1 is the Euler class of Q = V ⊖ S (we identify the cohomology of M ×V with the cohomology of M); and π_{∗} is integration along the fiber of π. □

Remark 3.2. If ϕ(X̃) has smaller dimension than X̃ then π_{∗}i_{∗}1 = ∫_{M} e(Q) is zero since this cohomology class is supported on ϕ(X̃) and its codimension is bigger than the rank of π_{∗}i_{∗}1.

Below we will calculate the integral (3.2) using the Berline-Vergne-Atiyah-Bott integral formula [?], [?], that we recall now.

Proposition 3.3 (Berline-Vergne-Atiyah-Bott). Suppose that M is a compact oriented manifold, T is a torus acting smoothly on M and C(M) is the set of components of the fixed point manifold. Then for any cohomology class α ∈ H_{T}^{∗} (M)

(3.3)  ∫_{M} α = ∑_{F ∈C(M)} ∫_{F} \frac{i_{F}^{∗} α}{e(ν_{F})}.

Here i_{F} : F → M is the inclusion, e(ν_{F}) is the T -equivariant Euler class of the normal bundle ν_{F} of F ⊂ M. The right side is considered in the fraction field of the polynomial ring of H_{T}^{∗} (point) = H^{∗}(BT) (see more on details in [?]): part of the statement is that the denominators cancel when the sum is simplified.

Remark 3.4. Here T denotes a real torus T = U(1)^{r}. Since BU(1)^{r} is homotopy equivalent to BGL(1)^{r} we can always restrict the complex torus action to the real one without losing any cohomological information.

Returning to the situation of Proposition 3.1, let us further assume that the group G is a torus, and that the fixed point set F (M) on the compact manifold M is finite. For a fixed point f ∈ F (M) the Euler class e(V) restricted to f can be computed as the product of the Euler classes of the restrictions of S and Q to f, that is e(V_{f}) = e(S_{f})e(Q_{f}). Also, ν_{f} is just the tangent space T_{f}M. Hence, from Proposition 3.1 and 3.3 we obtain that

(3.4)  [X] = e(V) ∑_{f∈F (M)} (e(S_{f}) · e(T_{f}M))^{−1} .

This formula is a special case of [?, Prop. 3.2]. These type of formulas have their origin in [?].

4. The equivariant cohomology class of the dual of smooth varieties

Let the torus T act on the complex vector space V and let X ⊂ V be a T -invariant cone. Assume moreover that ℙX ⊂ ℙV is smooth, and that the projective dual ℙX̌ of ℙX is a hypersurface. As a general reference we refer the reader to the book of Tevelev [?] on Projectively Dual Varieties. For a smooth ℙX ⊂ ℙV the projective dual of ℙX is simply the variety of projective hyperplanes tangent to ℙX.

Our objective in this section is to find a formula for the T -equivariant rational cohomology class of the dual X̌ of X ⊂ V, under certain conditions (X̌ is the cone of ℙX̌). The result will be a cohomology class in H^{∗}(BT) which we identify with the symmetric algebra of the character group of T. Since we use localization, our formulas will formally live in the fraction field of H^{∗}(BT), but part of the statements will be that in the outcome the denominators cancel.

Let V̌ denote the dual vector space of V and consider the conormal vector bundle

Ň = {(x, Λ) ∈ ℙX × V̌ | ℙΛ is tangent to ℙX at x},

and the diagram

\xymatrix{Ň \ar[rrr]^{ϕ} \ar@{^{(}->}[rr]^{i} \ar[dr] &  & ℙX × V̌ \ar[dl] \ar[r]^{π} & V̌ \\  & ℙX &  & X̌ \ar@{^{(}->}[u]}

(cf. Proposition 3.1).

One can check that the assumption that ℙX̌ is a hypersurface yields that the map ϕ is a resolution of X̌ (see [?, thm 1.10]). Hence from (3.4) we obtain the following statement.

Proposition 4.1. Assume that the fixed point set F (ℙX) is finite. Then the equivariant cohomology class represented by the cone X̌ is

[X̌] = ∑_{f∈F (ℙX)} \frac{e(V̌)}{e(T_{f}ℙX)e(Ň_{f}ℙX)}.

Let W(A) denote the set of weights of the T -module A counted with multiplicity. For a fixed point f ∈ F (ℙX) we have

W(V̌) = {−ω(f)} ∪ {−ω(f) − β|β ∈ W(T_{f}ℙX)} ∪ W(Ň_{f}ℙX),

where ω(f) denotes the weight corresponding to the fixed point f. Notice the asymmetrical role of Ň_{f}ℙX and T_{f}ℙX. The vector space Ň_{f}ℙX is a subspace of V̌ but T_{f}ℙX ⊂ T_{f}ℙV ≅ hom(L_{f}, V ⊖ L_{f}) where L_{f} ≤ V is the eigenline corresponding to the fixed point f. This isomorphism explains the shift by −ω(f) in the formula. Therefore we obtain

Theorem 4.2. Suppose that the torus T acts on the complex vector space V linearly and X ⊂ V is a T -invariant cone. Assume moreover that ℙX ⊂ ℙV is smooth with finitely many T -fixed points and the projective dual ℙX̌ is a hypersurface. Then the equivariant cohomology class represented by the cone X̌ of the dual ℙX̌ is

[X̌] = − ∑_{f∈F (ℙX)} ω(f) ∏_{β∈W(T_{f} ℙX)} \frac{β + ω(f)}{−β}.

Notice that the fixed points f of the T -action on ℙX correspond to eigenlines L_{f} ≤ V and the weight ω(f) of this line is canonically identified with the cohomology class e(L_{f}).

Remark 4.3. The difference of the dimensions of a hypersurface and the variety ℙX̌ is called the defect of ℙX. Hence Theorem 4.2 deals with the defect 0 case. It is customary to define the cohomology class of X̌ and the degree of ℙX̌ to be 0 if ℙX has positive defect. Using this convention Theorem 4.2 remains valid without the condition on the defect. Indeed, by Remark 3.2 the right hand side, which is equal to the pushforward of 1, is automatically zero if the image of π_{2} has smaller dimension.

Remark 4.4. Similar argument yields the cohomology class of ℙX itself: In Proposition 3.1 we choose S to be the tautological bundle over ℙX. As a result we obtain a nontrivial special case of the Duistermaat-Heckman formula: If the set F (ℙX) of fixed points on ℙX is finite, then

[X] = e(V) ∑_{f∈F (ℙX)}(ω(f) · e(T_{f}ℙX))^{−1}.

5. Cohomology and degree formulas for the discriminant

In this section we apply Theorem 4.2 to obtain formulas for the equivariant class and the degree of the discriminants of irreducible representations.

Let ρ : G → GL(V) be an irreducible representation of the complex connected reductive Lie group G on the complex vector space V, and let ℙX ⊂ ℙV be the (closed, smooth) orbit of [v] where v ∈ V is a vector corresponding to the highest weight λ. Let D_{λ} ⊂ V̌ and ℙD_{λ} ⊂ ℙV̌ be the duals of X and ℙX, they are called the “discriminants” of the representation of highest weight λ. The discriminant is “usually” a hypersurface, a complete list of representations of semisimple Lie groups for which the discriminant is not a hypersurfare (i.e. the defect is positive, c.f. Remark 4.3) can be found in [?], see also [?, Th.9.21].

Remark 5.1. For an irreducible representation of a reductive group G on V the action of the center of G is trivial on ℙV, that is, only the semisimple part of G acts on ℙV. Hence, seemingly it is enough to state the degree formula only for semisimple groups. However, we will calculate the degree of the discriminant from its equivariant Poicaré dual cohomology class, which is an element in H_{G}^{2}(point). For semisimple groups this cohomology group is 0. Therefore, if G is semisimple, we replace G with G × GL(1)—and let GL(1) act on V by multiplication. Consequently we are forced to state our theorems for reductive groups.

Let W be the Weyl group, let R(G) be the set of roots of G, and R^{−}(G) the set of negative roots. Let 𝔤_{β} be the root space corresponding to the root β. Let H_{β} be the unique element in [𝔤_{β}, 𝔤_{−β}] with β(H_{β}) = 2.

Theorem 5.2. Main Formula—Short Version. With G, W, V, λ, D_{λ}, R^{−}(G) as above, the equivariant Poincaré dual of D_{λ} in V̌ is

[D_{λ}] = − ∑_{μ∈W λ} μ ∏_{β∈T_{μ}} \frac{μ + β}{−β} ,

where

T_{λ} = {β ∈ R^{−}(G) | ⟨H_{β}, λ⟩ < 0},

and T_{wλ} = wT_{λ} for w ∈ W. Here we used the convention of Remark 4.3, i.e. the class of [D_{λ}] is defined to be 0 if D_{λ} is not a hypersurface.

Let us remark that {β ∈ R^{−}(G)|B(β, λ) < 0} is an equivalent description of T_{λ} (where B is the Killing form). The proof of Theorem 5.2 is based on the following two standard lemmas:

Lemma 5.3. The fixed point set F (ℙX) of the maximal torus T ⊂ G is equal to the orbit of [v] ∈ ℙV for the action of the Weyl group W.

Notice that the Weyl group W = N_{G}(T)/T indeed acts on ℙX since T fixes [v] ∈ ℙV.

Proof. It is enough to show that if [v] ∈ ℙV and g[v] are both fixed points of T and v is a maximal weight vector, then there exists a β ∈ N_{G}(T) such that g[v] = β[v].

Let G_{[v]} be the stabilizer of [v]. Then, by the assumption, T and g^{−1}T g are contained in G_{[v]}. These are maximal tori in G_{[v]}, so there is a p ∈ G_{[v]} such that g^{−1}T g = p^{−1}T p (see e.g. [?, p. 263]). Then β = gp^{−1} ∈ N_{G}(T) and g[v] = β[v]. □

Lemma 5.4. The weights of the tangent space T_{f}(ℙX) as a T -space are

T_{f} = {β ∈ R^{−}(G) | ⟨H_{β}, ω(f)⟩ < 0}

for any f ∈ F (ℙX).

For semisimple G the proof can be found in [?] and a more detailed version in [?, p.36]. The formula extends to the reductive case without change.

Proof of Theorem 5.2. It is enough to apply Theorem 4.2 to our situation. Lemma 5.3 determines the (finitely many) fixed points and Lemma 5.4 gives that

T_{λ} = {β ∈ R^{−}(G) | ⟨H_{β}, λ⟩ < 0}.

The weights at other fixed points are obtained by applying the appropriate element of the Weyl group. □

For the Lie group G = GL(n) the maximal torus can be identified with the subgroup of diagonal matrices diag(z_{1}, …, z_{n}), |z_{i}| = 1, which is the product of n copies of S^{1}’s. Let L_{i} be the character of this torus, which is the identity on the i’th S^{1} factor, and constant 1 on the others. The Weyl group, the symmetric group on n letters, permutes the L_{i}’s. The Weyl-invariant subring of the symmetric algebra ℚ[L_{1}, …, L_{n}] is identified with H^{∗}(BGL(n)). Under this identification, the i’th elementary symmetric polynomial of the L_{i}’s is the i’th Chern class (‘splitting lemma’). Hence—as usual—we call the L_{i}’s the Chern roots of GL(n).

Example 5.5. Consider the dual of Gr_{3}(ℂ^{8}) in its Plücker embedding. This is the discriminant of the representation of GL(8) with highest weight L_{1} + L_{2} + L_{3}. Let \binom{n}{k} stand for the set of k-element subsets of \{1, …, n\}. By Theorem 5.2 the class of the discriminant is

− ∑_{S∈\binom{8}{3}}(L_{s_{1}} + L_{s_{2}} + L_{s_{3}}) · ∏_{i∈S} ∏_{j∉S} \frac{L_{s_{1}} + L_{s_{2}} + L_{s_{3}} + L_{j} − L_{i}}{(L_{i} − L_{j})} .

This is a 58-term sum. However, this class is in H^{2}(BGL(8)), hence we know that it must have the form of v · c_{1} = v · ∑_{i=1}^{8} L_{i}. Therefore we only need to determine the number v. A well chosen substitution will kill most of the terms, making the calculation of v easier. E.g. substitute L_{i} = i − 13/3, then all terms—labeled by 3-element subsets S of {1, …, 8}—are zero, except for the last one corresponding to S = {6, 7, 8}. For the last term we obtain −8. Hence, −8 = v · (1 − 13/3 + 2 − 13/3 + … + 8 − 13/3), which gives the value of v = −6. Thus we obtain the equivariant class of the dual of Gr_{3}(ℂ^{8}) to be −6(L_{1} + … + L_{8}), and in turn its degree as −6(−1/3 − … − 1/3) = 16 (by Proposition 2.1).

Although similarly lucky substitutions cannot be expected in general, the L_{i} = i substitution yields a formula for the degree of the dual variety of the Grassmannian Gr_{k}(ℂ^{n}) in its Plücker embedding:

(5.1)  deg(Ǧr_{k}(ℂ^{n})) = \frac{2k}{n + 1} ∑_{S∈\binom{n}{k}} l(S) ∏_{i∈S,j∉S} \frac{l(S) + j − i}{i − j},

where l(S) = ∑_{s∈S} s.

To study how the degree depends on the highest weight λ for a fixed group G we introduce another expression for the degree where the sum is over all elements of the Weyl group instead of the orbit W λ.

For a dominant weight λ, let O_{λ} = R^{−}(G) \ T_{λ} = {β ∈ R^{−}(G) | ⟨H_{β}, λ⟩ = 0}, and let the sign ε(λ) of λ be (−1)^{|O_{λ}|}. Let W_{λ} ≤ W be the stabilizer subgroup of λ.

Theorem 5.6. Main Formula—Symmetric Version. Under the conditions of Theorem 5.2 we have

[D_{λ}] = −\frac{ε(λ)}{|W_{λ}|} ∑_{w∈W} w(λ ∏_{β∈R^{−}(G)} \frac{λ + β}{−β} ).

Proof. Let w_{1}, …, w_{m} be left coset representatives of W_{λ} ≤ W, i.e. the disjoint union of the w_{i}W_{λ}’s is W. Then

−\frac{ε(λ)}{|W_{λ}|} ∑_{w∈W} w (λ ∏_{β∈R^{−}(G)} \frac{λ + β}{−β} ) =

= −\frac{ε(λ)}{|W_{λ}|} ∑_{i=1}^{m} w_{i} ( ∑_{w∈W_{λ}} w (λ ∏_{β∈T_{λ}} \frac{λ + β}{−β} ∏_{β∈O_{λ}} \frac{λ + β}{−β} )) (5.2)

= −\frac{ε(λ)}{|W_{λ}|} ∑_{i=1}^{m} w_{i}(λ ∏_{β∈T_{λ}} \frac{λ + β}{−β} ∑_{w∈W_{λ}} w( ∏_{β∈O_{λ}} \frac{λ + β}{−β} )), (5.3)

since wT_{λ} = T_{λ} (but wO_{λ} is not necessarily equal to O_{λ}). Now we need the following lemma.

Lemma 5.7.

∑_{w∈W_{λ}} w( ∏_{β∈O_{λ}} \frac{λ + β}{−β} ) = ε(λ)|W_{λ}|.

Proof. The polynomial

P (x) = ∑_{w∈W_{λ}} ∏_{β∈O_{λ}} \frac{x + w(β)}{−w(β)} ∈ ℤ(L_{i})[x]

is W_{λ}-invariant, and has the form

\frac{q_{k} + q_{k−1}x + … + q_{0}x^{k}}{∏_{β∈O_{λ}} β}

where k = |O_{λ}| and q_{i} is a degree i polynomial on the orthogonal complement of λ. This means that the numerator has to be anti-symmetric under W_{λ} (which is itself a Weyl group of a root system), hence it must have degree at least the number of positive roots. That is, all q_{i}, i < k must vanish. This means that P (x) is independent of x, i.e. P (λ) = P (0) = ε(λ)|W_{λ}|, as required. □ Using this Lemma, formula (5.3) is further equal to

− ∑_{i=1}^{m} w_{i}(λ ∏_{β∈T_{λ}} \frac{λ + β}{−β} ),

which completes the proof of Theorem 5.6. □

The advantage of the Short Version (Theorem 5.2) is that for certain λ’s (λ’s on the boundary of the Weyl-chamber) the occurring products have only few factors, while the advantage of the Symmetric Version (Theorem 5.6) is that it gives a unified formula for all λ’s of a fixed group. Now we will expand this latter observation.

Let G be semisimple and consider the representation with highest weight λ. Extend this action to an action of G × GL(1) with GL(1) acting by scalar multiplication. Denoting the first Chern class of GL(1) by u (that is H^{∗}(BU(1)) = ℚ[u]) we obtain that

(5.4)  [D_{λ}] = −\frac{ε(λ)}{|W_{λ}|} ∑_{w∈W} w((λ + u) ∏_{β∈R^{−}(G)} \frac{λ + u + β}{−β} ).

Proposition 2.1 then turns this formula to a degree formula for the discriminant, by substituting u = −1:

(5.5)  deg(D_{λ}) = −\frac{ε(λ)}{|W_{λ}|} ∑_{w∈W} w((λ − 1) ∏_{β∈R^{−}(G)} \frac{λ + β − 1}{−β}).

Part of the statement is that this formula is a constant, i.e. expression (5.4) is equal to a constant times u (although this can also be deduced from the fact that H^{2}(BG) = 0 for semisimple groups).

Corollary 5.8. Let G be a semisimple Lie group, and 𝔥 the corresponding Cartan subalgebra. There exists a polynomial F_{G} : 𝔥^{∗} → ℤ, with degree equal the number of positive roots of G, such that

deg(D_{λ}) = \frac{ε(λ)}{|W_{λ}|}F_{G}(λ)

if λ is a dominant weight. The polynomial F_{G}(λ) vanishes if and only if D_{λ} is not a hypersurface.

Proof. We have

(5.6)  F_{G} = − ∑_{w∈W} w((λ − 1) ∏_{β∈R^{−}(G)} \frac{λ + β − 1}{−β}),

and the last statement follows from Remark 4.3. □

Remark 5.9. The polynomial dependence of deg(D_{λ}) for regular weights λ (hence ε(λ) = 1, W_{λ} = {1}), as well as a formula for a special value of the polynomial (the value at the sum of the fundamental weights) is given in [?]. Since deg(D_{λ}) is always non-negative the value of ε(λ) is determined by the sign of F_{G}(λ). The positive defect cases are known but this corollary provides an alternative and uniform way to find them.

Example 5.10. A choice of simple roots α_{1}, α_{2}, …, α_{r} of G determines the fundamental weights ω_{1}, ω_{2}, …, ω_{r} by B(ω_{i}, \frac{2α_{j}}{B(α_{j},α_{j})}) = δ_{i,j} where B(., .) denotes the Killing form. We follow the convention of De Concini and Weyman [?] by writing F_{G} in the basis of fundamental weights (i.e. y_{1}ω_{1} + … + y_{r}ω_{r} ↦ F_{G}(y_{1}, …, y_{r})) and substituting y_{i} = x_{i} + 1. The advantage of this substitution is that in this way, according to [?], all the coefficients of the polynomial F_{G} become non-negative. Formula (5.6) gives the following polynomials for all semisimple Lie groups of rank at most 2 and for some of rank 3. [In these examples our convention for simple roots agrees with the one in the coxeter/weyl Maple package www.math.lsa.umich.edu/˜ jrs/maple.html written by J. Stembridge.]

A_{1}, α_{1} = L_{2} − L_{1}:

F = 2x_{1}.

A_{1} + A_{1}, α = (L_{2} − L_{1}, L′_{2} − L′_{1}):

F = 6x_{1}x_{2} + 2x_{2} + 2x_{1} + 2.

A_{2}, α = (L_{2} − L_{1}, L_{3} − L_{2}):

F = 6(x_{1} + x_{2} + 1)(2x_{1}x_{2} + x_{1} + x_{2} + 1) [?, ex 7.18].

B_{2}, α = (L_{1}, L_{2} − L_{1}),

F = 20(2x_{2}^{3}x_{1} +3x_{2}^{2}x_{1}^{2} +x_{2}x_{1}^{3})+12(2x_{2}^{3} +12x_{2}^{2}x_{1} +11x_{2}x_{1}^{2} +x_{1}^{3})+24(3x_{2}^{2} + 7x_{2}x_{1} + 2x_{1}^{2}) + 8(9x_{2} + 8x_{1}) + 24.

G_{2}, α = (L_{2} − L_{1}, L_{1} − 2L_{2} + L_{3}):

F = 42(18x_{2}^{5}x_{1} + 45x_{2}^{4}x_{1}^{2} + 40x_{2}^{3}x_{1}^{3} + 15x_{2}^{2}x_{1}^{4} + 2x_{2}x_{1}^{5}) + 60(9x_{2}^{5} + 90x_{2}^{4}x_{1} + 150x_{2}^{3}x_{1}^{2} + 90x_{2}^{2}x_{1}^{3} + 20x_{2}x_{1}^{4} + x_{1}^{5}) + 110(27x_{2}^{4} + 132x_{2}^{3}x_{1} + 144x_{2}^{2}x_{1}^{2} + 52x_{2}x_{1}^{3} + 5x_{1}^{4})+8(822x_{2}^{3} +2349x_{2}^{2}x_{1} +1527x_{2}x_{1}^{2} +248x_{1}^{3})+6(60x_{2}^{2} +1972x_{2}x_{1} +579x_{1}^{2})+ 4(1025x_{2} + 727x_{1}) + 916.

A_{1} + A_{1} + A_{1}, α = (L_{2} − L_{1}, L′_{2} − L′_{1}, L′′_{2} − L′′_{1}):

F = 24x_{2}x_{1}x_{3} + 12(x_{2}x_{1} + x_{2}x_{3} + x_{1}x_{3}) + 8(x_{2} + x_{1} + x_{3}) + 4.

A_{1} + A_{2}, α = (L_{2} − L_{1}, L′_{2} − L′_{1}, L′_{3} − L′_{2}):

F = 60x_{2}^{2}x_{1}x_{3} + 60x_{2}x_{1}x_{3}^{2} + 36x_{2}^{2}x_{1} + 36x_{2}^{2}x_{3} + 144x_{2}x_{1}x_{3} + 36x_{2}x_{3}^{2} + 36x_{1}x_{3}^{2} + 24x_{2}^{2} + 72x_{2}x_{1} + 96x_{2}x_{3} + 72x_{1}x_{3} + 24x_{3}^{2} + 48x_{2} + 36x_{1} + 48x_{3} + 24.

A_{3}, α = (L_{2} − L_{1}, L_{3} − L_{2}, L_{4} − L_{3}):

F = 420x_{1}x_{3}x_{2}(x_{1} + x_{2})(x_{2} + x_{3})(x_{2} + x_{1} + x_{3}) + 300(x_{1}^{3}x_{2}^{2} + 4x_{1}^{3}x_{2}x_{3} + x_{1}^{3}x_{3}^{2} + 2x_{1}^{2}x_{2}^{3} + 15x_{1}^{2}x_{2}^{2}x_{3} + 12x_{1}^{2}x_{2}x_{3}^{2} + x_{1}^{2}x_{3}^{3} + x_{1}x_{2}^{4} + 12x_{1}x_{2}^{3}x_{3} + 15x_{1}x_{2}^{2}x_{3}^{2} + 4x_{1}x_{2}x_{3}^{3} + x_{2}^{4}x_{3} + 2x_{2}^{3}x_{3}^{2} + x_{2}^{2}x_{3}^{3}) + 220(3x_{1}^{3}x_{2} + 3x_{1}^{3}x_{3} + 12x_{1}^{2}x_{2}^{2} + 33x_{1}^{2}x_{2}x_{3} + 9x_{1}^{2}x_{3}^{2} + 10x_{1}x_{2}^{3} + 48x_{1}x_{2}^{2}x_{3} + 33x_{1}x_{2}x_{3}^{2} + 3x_{1}x_{3}^{3} + x_{2}^{4} + 10x_{2}^{3}x_{3} + 12x_{2}^{2}x_{3}^{2} + 3x_{2}x_{3}^{3}) + 8(42x_{1}^{3} + 453x_{1}^{2}x_{2} + 411x_{1}^{2}x_{3} + 699x_{1}x_{2}^{2} + 1566x_{1}x_{2}x_{3} + 411x_{1}x_{3}^{2} + 164x_{2}^{3} +699x_{2}^{2}x_{3} +453x_{2}x_{3}^{2} +42x_{3}^{3})+12(126x_{1}^{2} +491x_{1}x_{2} +407x_{1}x_{3} +239x_{2}^{2} + 491x_{2}x_{3} + 126x_{3}^{2}) + 16(133x_{1} + 169x_{2} + 133x_{3}) + 904.

A Maple computer program computing F for any semisimple Lie group is available at www.unc.edu/˜ rimanyi/progs/feherpolinom.mw.

Straightforward calculation gives that F_{G} for G = A_{1} + … + A_{1} (n times) is

F_{nA_{1}}(y_{1}, …, y_{n}) = ∑_{k=0}^{n}(−2)^{n−k}(k + 1)! σ_{k}(y_{1}, …, y_{n}),

where σ_{i} is the i’th elementary symmetric polynomial. This can also be derived from [?, Th.2.5,Ch.13].

6. Combinatorics of the degree formulas for GL(n)

Our main formulas, Theorems 5.2 and 5.6, can be encoded using standard notions from the combinatorics of symmetric functions. In Sections 6-7 we assume that G = GL(n), and that the simple roots are L_{i} − L_{i+1}. Then R^{−}(G) = {L_{i} − L_{j} : i > j}. Irreducible representations correspond to the weights λ = ∑_{i=1}^{n} a_{i}L_{i} with a_{1} ≥ a_{2} ≥ ⋯ ≥ a_{n}. For such a weight one has T_{λ} = {L_{i} − L_{j} : i > j, a_{i} < a_{j}}.

6.1. Symmetrizer operators. Let

λ^{+} = λ ∏_{β∈R^{−}(G)}(λ + β),  and  Δ = ∏_{1≤i<j≤n}(L_{i} − L_{j}),

and recall the definition of the Jacobi-symmetrizer ([?]) of a polynomial f(L_{1}, …, L_{n}):

J(f)(L_{1}, …, L_{n}) = \frac{1}{Δ} ∑_{w∈S_{n}} sgn(w)f(L_{w(1)}, …, L_{w(n)}),

where sgn(w) is the sign of the permutation w, i.e. (−1) raised to the power of the number of transpositions in w. Then for λ = λ_{1}L_{1} + … + λ_{n}L_{n} with λ_{1} ≥ λ_{2} ≥ … ≥ λ_{n}, from Theorem 5.6 we obtain

(6.1)  [D_{λ}] = −\frac{ε(λ)}{|W_{λ}|}J(λ^{+}),

and deg D_{λ} is obtained by multiplying this by the factor

(6.2)  \frac{−n}{|λ| · σ_{1}} , where |λ| := ∑ λ_{i} and σ_{1} := ∑ L_{i}.

The Jacobi symmetrizer is a special case of the divided difference operators ∂_{w} [?, Ch.7], corresponding to the maximal permutation w_{0} = [n, n − 1, …, 2, 1] (i.e. w_{0}(i) = n + 1 − i). Shorter divided difference operators also turn up for certain representations. We will illustrate this with the case of the Plücker embedding of Grassmann varieties, i.e. G = GL(n), λ = L_{1}+…+L_{k}. Let [k|n−k] ∈ S_{n} be the permutation [n−k+1, n−k+2, …, n, 1, 2, …, n−k]. Theorem 5.2 gives

[D_{L_{1}+...+L_{k}}] = −∂_{[k|n−k]}((L_{1} + … + L_{k}) ∏_{i=1}^{k} ∏_{j=k+1}^{n}(L_{1} + … + L_{k} + L_{j} − L_{i})),

and the degree of the discriminant of Gr_{k}(ℂ^{n}) in its Plücker embedding is obtained by multiplying this with −n/(kσ_{1}).

6.2. Scalar product. Now we show how to use the scalar product on function spaces to encode the formula of Theorem 5.6 in the case of G = GL(n). Following [?], we define the scalar product of the polynomials f, g in n variables L_{1}, …, L_{n} as

⟨f, g⟩ = \frac{1}{n!}[fḡ ∏_{i≠j} (1 − \frac{L_{i}}{L_{j}} )]_{1},

where ḡ(L_{1}, …, L_{n}) = g(1/L_{1}, …, 1/L_{n}) and [h]_{1} is the constant term (i.e. the coefficient of 1) of the Laurent polynomial h ∈ ℤ[L_{1}^{±}, …, L_{n}^{±}]. The Jacobi symmetrizer is basically a projection, thus for a degree \binom{n}{2} + 1 polynomial f we have

\frac{1}{n!} ∑_{w} sgn(w)f(L_{w(1)}, …, L_{w(n)}) = \frac{⟨f, σ_{1}Δ⟩}{⟨σ_{1}Δ, σ_{1}Δ⟩}σ_{1}Δ.

Here ⟨σ_{1}Δ, σ_{1}Δ⟩ can be calculated to be n(2n−3)!!, where (2k+1)!! denotes the semifactorial 1 · 3 ⋯ (2k − 1) · (2k + 1), hence we obtain

[D_{λ}] = \frac{−ε(λ)n!}{|W_{λ}| · n(2n − 3)!!}⟨λ^{+}, σ_{1}Δ⟩σ_{1},

and hence the following form of our Main Formula:

Theorem 6.1. The degree of the discriminant of the irreducible representation of GL(n) with highest weight λ is

deg D_{λ} = \frac{ε(λ)}{|λ| · |W_{λ}|} \frac{n!}{(2n − 3)!!}⟨λ^{+}, σ_{1}Δ⟩.

6.3. Permanent. For (ν_{1}, …, ν_{n}) ∈ ℕ^{n} and w ∈ S_{n} let w(ν) = (ν_{w(1)}, …, ν_{w(n)}) and L^{ν} will denote the monomial L_{1}^{ν_{1}} … L_{n}^{ν_{n}}.

Lemma 6.2. Let μ = (n, n − 2, n − 3, …, 2, 1, 0) ∈ ℕ^{n}. If ∑ ν_{i} = \binom{n}{2} + 1 then we have

J(L^{ν}) = \begin{cases}sgn(w)σ_{1} & if ν_{i} = μ_{w(i)} \\ 0 & otherwise \end{cases}

Proof. If ν_{i} = ν_{j} then the terms of J(L^{ν}) turn up in cancelling pairs, hence J(L^{ν}) = 0. This leaves only ν = permutations of μ for possible nonzero J-value. Direct check shows J(L^{μ}) = σ_{1} (c.f. the well known identity J (L^{(n−1,n−2,...,2,1,0)}) = 1). □

The coefficient of a monomial L^{ν} in a polynomial f will be denoted by c(f, L^{ν}). Formula (6.1) and Lemma 6.2 yield that the class and degree of D_{λ} can be computed by counting coefficients.

Theorem 6.3. The equivariant class of D_{λ} is

[D_{λ}] = −\frac{ε(λ)}{|W_{λ}|} ∑_{w∈S_{n}} sgn(w)c(λ^{+}, L^{w(μ)}) · σ_{1},

and the degree of D_{λ} is obtained by multiplication by −n/(|λ|σ_{1}).

Similar sums will appear later, hence we define the ν-permanent of a polynomial f as

∑_{w∈S_{n}} sgn(w)c(f, L^{w(ν)}),

and denote it by P (f, ν).

The name permanent is justified by the following observation. Let ν = (ν_{1}, …, ν_{n}) be a partition. If the polynomial f is the product of |ν| linear factors ∑_{j=1}^{n} a_{j}^{(i)}L_{j}, i = 1, …, |ν|, then P (f, ν) can be computed from the n × |ν| matrix (a_{j}^{(i)}) as follows. For a permutation w ∈ S_{n} we choose ν_{1} entries from row ν(1) (i.e. j = ν(1)), ν_{2} entries from row ν(2), etc, such a way that no chosen entries are in the same column (hence they form a complete ‘rook arrangement’). The product of the chosen entries with sign sgn(w) will be a term in P (f, ν), and we take the sum for all w ∈ S_{n} and all choices. For example the (2, 1)-permanent of the product (aL_{1} + a′L_{2})(bL_{1} + b′L_{2})(cL_{1} +

c′L_{2}), considering the matrix (\begin{matrix}a & b & c \\ a′ & b′ & c′ \end{matrix}), is

abc′ + ab′c + a′bc − a′b′c − a′bc′ − ab′c′.

Theorem 6.4. (Boole’s formula, [?, 7.1]) If λ = aL_{1}, then the degree of D_{λ} is n(a − 1)^{n−1}.

Proof. We have

λ^{+} = aL_{1} ∏_{i=2}^{n} ((a − 1)L_{1} + L_{i}) · ∏_{2≤i<j≤n}(L_{j} − L_{i} + aL_{1}),

and let

λ^{++} = aL_{1} ∏_{i=2}^{n}((a − 1)L_{1} + L_{i}) · ∏_{2≤i<j≤n}(L_{j} − L_{i}).

Observe that P (λ^{+}, μ) = P (λ^{++}, μ), since in the difference each term comes once with a positive, once with a negative sign. In P (λ^{++}, μ) only L_{1} has degree ≥ n (namely n), hence we have

P (λ^{++}, μ) = a(a − 1)^{n−1}P ( ∏_{2≤i<j≤n}(L_{i} − L_{j}), (n − 2, n − 3, …, 2, 1))

which is further equal to a(a − 1)^{n−1}(−1)^{\binom{n−1}{2}}(n − 1)! and Theorem 6.3 gives the result. □

6.4. Hyperdeterminants. The discriminant of the standard action of the product group ∏_{u=1}^{k} GL(n_{u}) on ⨂_{u=1}^{k} ℂ^{n_{u}} is called hyperdeterminant because it generalizes the case of the determinant for k = 2. In other words, the hyperdeterminant is the dual of the Segre embedding ∏_{u=1}^{k} ℙ(ℂ^{n_{u}}) → ℙ(⨂_{u=1}^{k} ℂ^{n_{u}}). Gelfand, Kapranov and Zelevinsky [?] give a concrete description of the degree of the hyperdeterminants by giving a generator function. In the so-called boundary case when n_{k} − 1 = ∑_{u=1}^{k−1}(n_{u} − 1) they get a closed formula. We show how to prove this using ν-permanents.

When considering representations of ∏_{u=1}^{k} GL(n_{u}) (n_{1} ≤ n_{2} ≤ … ≤ n_{k}) we will need k sets of variables (the k sets of Chern roots), we will call them L_{(u),i}, u = 1, …, k, i = 1, …, n_{u}. The hyperdeterminant is the discriminant corresponding to the representation with highest weight λ = ∑_{u=1}^{k} L_{(u),1}. If ν_{(u)} ∈ ℕ^{k_{u}}, then a polynomial f in these variables has a ν = (ν_{(1)}, …, ν_{(k)})- permanent, defined as

P (f, ν) = ∑_{w^{(1)}∈S_{n_{1}}} … ∑_{w^{(k)}∈S_{n_{k}}} ( ∏_{u=1}^{k} sgn(w^{(u)}))c(f, ∏_{u} L_{(u)}^{w^{(u)}(ν_{(u)})}).

Theorem 6.5 ([?, 14.2.B]). If n_{k} − 1 > ∑_{u=1}^{k−1}(n_{u} − 1) then the discriminant is not a hypersurface. If n_{k} − 1 = ∑_{u=1}^{k−1}(n_{u} − 1) then its degree is n_{k}!/ ∏_{u=1}^{k−1}(n_{u} − 1)!.

Proof. Let μ^{(u)} = (μ_{(1)}, …, μ_{(k)}), where μ_{(v)} = (n_{v} − 1, n_{v} − 2, …, 1, 0) for v ≠ u, and μ_{(u)} = (n_{u}, n_{u} − 2, n_{u} − 3, …, 2, 1, 0). A straightforward generalization of the k = 1 case gives

(6.3)  [D_{λ}] = −\frac{(−1)∑^{\binom{nu−1}{2}}}{∏_{u=1}^{k}(n_{u} − 1)!} ∑_{u=1}^{k} (P (λ^{+}, μ^{(u)}) ∑_{i=1}^{n_{u}} L_{(u),i}),

where

λ^{+} = λ (∏_{u=1}^{k} ∏_{i=2}^{n_{u}}(L_{(u),i} − L_{(u),1} + λ)) (∏_{u=1}^{k} ∏_{2≤i<j≤n_{u}}(L_{(u),j} − L_{(u),i} + λ)) .

Just like in the proof of the Boole formula above, we can change λ^{+} to

λ^{++} = λ (∏_{u=1}^{k} ∏_{i=2}^{n_{u}}(L_{(u),i} − L_{(u),1} + λ)) (∏_{u=1}^{k} ∏_{2≤i<j≤n_{u}}(L_{(u),j} − L_{(u),i}))

without changing the value of the permanent. Observe that the highest power of an L_{(k),i} variable in λ^{++} is ∑_{u=1}^{k−1}(n_{u} − 1), thus the last term in (6.3) is 0 if n_{k} − 1 > ∑_{u=1}^{k−1}(n_{u} − 1). This proves the first statement of the theorem.

If we have n_{k} − 1 = ∑_{u=1}^{k−1}(n_{u} − 1), then considering the power of L_{(k),1} again, we get that

P (λ^{++}, μ_{(k)}) =

P (( ∑_{u=1}^{k−1} L_{(u),1})^{n_{k}−1} ∏_{u=1}^{k} ∏_{2≤i<j≤n_{u}}(L_{(u),j} − L_{(u),i}), ((n_{1} − 1, …, 1, 0), …

…, (n_{k−1} − 1, …, 1, 0), (n_{k} − 2, …, 1, 0))) =

\frac{(n_{k} − 1)!}{∏_{u=1}^{k−1}(n_{u} − 1)!}P ( ∏_{u=1}^{k} ∏_{2≤i<j≤n_{u}}(L_{(u),j}−L_{(u),i}), ((n_{1}−2, …, 1, 0), …, (n_{k}−2, …, 1, 0))) =

(−1)∑^{\binom{nu−1}{2}} \frac{(n_{k} − 1)!}{∏_{u=1}^{k−1}(n_{u} − 1)!} ∏_{u=1}^{k}(n_{u} − 1)! = (−1)∑^{\binom{nu−1}{2}}(n_{k} − 1)!^{2}.

Hence, in the expansion −[D_{λ}] = c_{1} ∑ L_{(1),i} +c_{2} ∑ L_{(2),i} +…+c_{k} ∑ L_{(k),i} we have

c_{k} = (n_{k} − 1)!^{2}/ ∏_{u=1}^{k}(n_{u} − 1)! = (n_{k} − 1)!/ ∏_{u=1}^{k−1}(n_{u} − 1)!

Proposition 2.1 then gives deg(D_{λ}) by substituting (e.g.) L_{(u),i} = 0 for u < k and L_{(k),i} = −1 into [D_{λ}], which proves the Theorem. □

7. Some more explicit formulae for GL(n).

Notice that the expression of [D_{λ}] from 5.2 is homogeneous in L_{1}, …, L_{n} of degree 1. Moreover, [D_{λ}] · Δ is anti-symmetric of degree deg Δ + 1, hence it is the product of Δ and a symmetric polynomial of degree 1. Hence, by (6.2), for some constant f_{λ}^{(n)} one has

(7.1)  [D_{λ}] = −f_{λ}^{(n)} · σ_{1} and deg(D_{λ}) = \frac{n}{|λ|} · f_{λ}^{(n)},

where σ_{1} = ∑_{i=1}^{n} L_{i} as above. By similar argument one deduces that for a free variable t one has

R_{λ}^{(n)}(t) := ∑_{μ∈W λ}(μ + t) ∏_{β∈T_{μ}} \frac{μ + t + β}{−β} = A_{λ}^{(n)}t + f_{λ}^{(n)}σ_{1}

for another constant A_{λ}^{(n)}. Since the dual variety associated with the representation ρ_{λ} and the dual variety associated with the action ρ_{λ} × diag of GL(n) × GL(1) are the same, their degrees are the same too. Applying our main result for these two groups and actions, one gets

(7.2)  R_{λ}^{(n)}(t) = f_{λ}^{(n)} · (\frac{n}{|λ|} · t + σ_{1}).

In fact, (7.2) is an entirely algebraic identity, and it is valid for any weight λ (i.e. not only for the dominant weights).

Using (7.2), we prove some identities connecting different weights.

Lemma 7.1. For any λ = ∑_{i} a_{i}L_{i} define \overline{λ} := ∑_{i} a_{i}L_{n−i}. Assume that either (i) λ′ = λ + aσ_{1}, or (ii) \overline{λ′} + λ = aσ_{1} for some a ∈ ℤ. Then

deg(D_{λ′}) = deg(D_{λ}).

Proof. In case (i), using (7.2), one gets T_{λ+aσ_{1}} = T_{λ} and f_{λ+aσ_{1}}^{(n)} = f_{λ}^{(n)}(\frac{na}{|λ|}+ 1), hence f_{λ+aσ_{1}}^{(n)}/|λ + aσ_{1}| = f_{λ}^{(n)}/|λ|. Then apply (7.1). For (ii) notice that T_{λ′} = \{−\overline{β} : β ∈ T_{λ}\}, hence −[D_{λ′}] equals

∑_{μ∈W λ} \overline{aσ_{1} − λ} ∏_{β∈T_{μ}} \frac{\overline{aσ_{1} − λ} + \overline{−β}}{\overline{β}} = − ∑_{μ∈W λ}(λ − aσ_{1}) ∏_{β∈T_{μ}} \frac{λ − aσ_{1} + β}{−β} =

−f_{λ}^{(n)}( − \frac{na}{|λ|} + 1)σ_{1}.

Therefore, f_{λ′}^{(n)} = f_{λ}^{(n)}(\frac{na}{|λ|} − 1), or f_{λ′}^{(n)}/|λ′| = f_{λ}^{(n)}/|λ|.

Geometrically the lemma corresponds to the fact that tensoring with the one dimensional representation or taking the dual representation does not change the discriminant. □ It follows from the lemma that we can always assume that the last coefficient a_{n} of λ is zero.

Example 7.2. The case of λ = (a + b)L_{1} + b(L_{2} + ⋯ + L_{n−1}) with a, b > 0 and n ≥ 3. Set λ′ := (a + b)L_{1} + a(L_{2} + ⋯ + L_{n−1}). Then \overline{λ′} + λ = (a + b)σ_{1}, hence deg(D_{λ}) = deg(D_{λ′}) by 7.1. In particular, deg(D_{λ}) is a symmetric polynomial in variables (a, b). In the sequel we deduce its explicit form. For this notice that

T_{λ} = {L_{i} − L_{1}, L_{n} − L_{i} : 2 ≤ i ≤ n − 1} ∪ {L_{n} − L_{1}}.

It is convenient to write λ as aL_{1} −bL_{n} +bσ_{1}. First, assume that a ≠ b. Using (7.1) and Lemma 7.1 one gets

deg(D_{λ}) = \frac{n}{|λ|}f_{λ}^{(n)} = \frac{n}{a + b(n − 1)}f_{aL_{1}−bL_{n}}^{(n)} · (\frac{nb}{a − b} + 1) = \frac{n}{a − b} · f_{aL_{1}−bL_{n}}^{(n)},

where f_{aL_{1}−bL_{n}}^{(n)} · σ_{1} equals (7.3)

∑_{i≠j}(aL_{i}−bL_{j})\frac{aL_{i} − bL_{j} + L_{j} − L_{i}}{L_{i} − L_{j}} ∏_{k≠i,j} \frac{aL_{i} − bL_{j} + L_{k} − L_{i}}{L_{i} − L_{k}} ·\frac{aL_{i} − bL_{j} + L_{j} − L_{k}}{L_{k} − L_{j}} .

Clearly, f_{aL_{1}−bL_{n}}^{(n)} = lim_{L_{n}→∞} (f_{aL_{1}−bL_{n}}^{(n)} · σ_{1})/L_{n}. (In the sequel we write simply lim for lim_{L_{n}→∞}.) In order to determine this limit, we separate L_{n} in the expression E of (7.3). We get three types of contributions: E = I +II +III, where I contains those terms of the sum ∑_{i≠j} where i, j ≤ n−1; II contains the terms with i = n; while III those terms with j = n.

One can see easily that lim I is finite (for any fixed L_{1}, …, L_{n−1}), hence lim I/L_{n} = 0.

The second term (after re-grouping) is

II = \frac{S}{∏_{j≤n−1} L_{n} − L_{k}} ,

where

S := ∑_{j≤n−1} \frac{∏_{k≤n−1}(aL_{n} − bL_{j} − L_{n} + L_{k})(aL_{n} − bL_{j} + L_{j} − L_{k})}{∏_{n≠k≠j} L_{k} − L_{j}}.

By similar argument as above, S can be written as S = ∑_{i=0}^{n} P_{i} · L_{n}^{n−i}, where P_{i} is a symmetric polynomial in L_{1}, …, L_{n−1} of degree i (and clearly also depends on a, b and n). In particular, lim II/L_{n} is the ‘constant’ P_{0} = P_{0}(a, b, n).

Set t_{1} := aL_{n}−bL_{j} −L_{n} and t_{2} := aL_{n}−bL_{j} +L_{j}. Notice that if we modify the polynomial P = ∏_{k≤n−1}(t_{1} + L_{k})(t_{2} − L_{k}) by any polynomial situated in the ideal generated by the symmetric polynomials σ_{1}^{(n−1)} := ∑_{i−1}^{n−1} L_{i}, σ_{2}^{(n−1)}, … (in L_{1}, …, L_{n−1}), then the modification has no effect in the ‘constant’ term P_{0}. In particular, since P = (t_{1}^{n−1} + σ_{1}^{(n−1)}t_{1}^{n−2} + ⋯)(t_{2}^{n−1} − σ_{1}^{(n−1)}t_{2}^{n−2} + ⋯), in the expression of S, the polynomial P can be replaced by t_{1}^{n−1}t_{2}^{n−1}. Therefore, if [Q(t)]_{t^{i}} denotes the coefficient of t^{i} in Q(t), then one has:

P_{0}(a, b, n) = [ ∑_{j≤n−1} \frac{(aL_{n} − bL_{j} − L_{n})^{n−1}(aL_{n} − bL_{j} + L_{j})^{n−1}}{∏_{n≠k≠j} L_{k} − L_{j}} ]_{L_{n}^{n}} .

Set

P_{a,b}(t) = At^{2} + Bt + C := (at − t − b)(at − b + 1).

Then

P_{0}(a, b, n) = [P_{a,b}(t)^{n−1}]_{t^{n}} · ∑_{j≤n−1} \frac{L_{j}^{n−2}}{∏_{n≠k≠j} L_{k} − L_{j}} .

Notice that the last sum is exactly (−1)^{n−2} by Lagrange interpolation formula.

A very similar computation provides lim III/L_{n}, and using the symmetry of P_{a,b} one finds that

deg(D_{λ}) = \frac{(−1)^{n}n}{a − b} ([P_{a,b}(t)^{n−1}]_{t^{n}} − [P_{a,b}(t)^{n−1}]_{t^{n−2}}) =

\frac{(−1)^{n}n}{a − b} ([P_{a,b}(t)^{n−1}]_{t^{n}} − [P_{b,a}(t)^{n−1}]_{t^{n}}) .

If ∂ denotes the ‘divided difference’ operator ∂Q(a, b) := (Q(a, b)−Q(b, a))/(a− b), then the last expression reads as

deg(D_{λ}) = (−1)^{n}n · ∂[P_{a,b}(t)^{n−1}]_{t^{n}}.

By Corollary 5.8, this is valid for a = b as well. By a computation (using the multinomial formula):

(7.4)  deg(D_{λ}) = n!(a + b − 1) ∑_{i=1}^{[n/2]} \frac{A^{i−1}C^{i−1}(−B)^{n−2i}}{i!(i − 1)!(n − 2i)!} .

This has the following factorization:

deg(D_{λ}) = n(n − 1)(a + b − 1)(−B)^{n−2[n/2]} · ∏_{i=1}^{[n/2]−1}(B^{2} + ξ_{i}AC),

where

∏_{i=1}^{[n/2]−1}(t + ξ_{i}) = t^{[n/2]−1} + \frac{(n − 2)!}{2!1!(n − 4)!}t^{[n/2]−2} + \frac{(n − 2)!}{3!2!(n − 6)!}t^{[n/2]−3} + · · · .

E.g., for small values of n one has the following expressions for deg(D_{λ}):

\begin{matrix}n = 3 & 6(a + b − 1)(−B) \\ n = 4 & 12(a + b − 1)(B^{2} + AC) \\ n = 5 & 20(a + b − 1)(−B)(B^{2} + 3AC) \\ n = 6 & 30(a + b − 1)(B^{2} + ξAC)(B^{2} + ξ̄AC), where ξ^{2} + 6ξ + 2 = 0. \end{matrix}

For n = 3, deg(D_{λ}) = 6(a + b − 1)(2ab − a − b + 1) is in fact the universal polynomial F_{GL(3)} (which already was determined in Example 5.10 case A_{2}, and from which one can determine the degrees of all the irreducible GL(3)- representations by Corollary 5.8), compare also with [?][7.18].

If we write u := a − 1 and v := b − 1, then all the coefficients of the polynomials −B, A, C, expressed in the new variables (u, v), are non-negative. Using (7.4), this remains true for deg(D_{λ}) as well; a fact compatible with [?].

For a = 1 the formula simplifies drastically (since A = 0): deg(D_{λ}) = n(n− 1)b^{n−1}. Symmetrically for b = 1. For a = b = 1 we recover the well known formula deg(D_{λ}) = n(n − 1) for the degree of the discriminant of the adjoint representation. (A matrix is in the discriminant of the adjoint representation if it has multiple eigenvalues i.e. the equation of the discriminant is the discriminant of the characteristic polynomial.)

Example 7.3. The case of λ = aL_{1} + bL_{2} with a > b ≥ 1 and n ≥ 3. Since T_{λ} = {L_{i} − L_{1}, L_{i} − L_{2} : 3 ≤ i ≤ n} ∪ {L_{2} − L_{1}}, one has the following expression for f_{λ}^{(n)} · σ_{1}: (7.5)

∑_{i≠j}(aL_{i}+bL_{j})\frac{aL_{i} + bL_{j} + L_{j} − L_{i}}{L_{i} − L_{j}} ∏_{k≠i,j} \frac{aL_{i} + bL_{j} + L_{k} − L_{i}}{L_{i} − L_{k}} ·\frac{aL_{i} + bL_{j} + L_{k} − L_{j}}{L_{j} − L_{k}} .

We compute f_{λ}^{(n)} as in 7.2. For this we write the expression E from (7.5) as I + II + III, where I, II and III are defined similarly as in 7.2. The computations of the first two contributions are similar as in 7.2: lim I/L_{n} = 0 while

lim II/L_{n} = [(at − t + b)^{n−1}(at + b − 1)^{n−1}]_{t^{n}}.

On the other hand, III is slightly more complicated. For this write t_{1} := aL_{i} + bL_{n} − L_{i}, t_{2} := aL_{i} + bL_{n} − L_{n}, and Π := ∏_{n≠k≠i} L_{i} − L_{k}. Moreover, we define the relation R_{1} ≡ R_{2} whenever lim R_{1}/L_{n} = lim R_{2}/L_{n}. Then (for t_{1} using the usual ‘trick’ as above by separating terms from the ideal I_{sym} generated by σ_{i}^{(n−1)}’s) we have that III =

= ∏_{k≤n−1} \frac{1}{L_{n} − L_{k}} · ∑_{i≤n−1} \frac{∏_{k≤n−1}(t_{1} + L_{k}) · (aL_{i} + bL_{n} + L_{n} − L_{i}) · ∏_{n≠k≠i}(t_{2} + L_{k})}{Π}

≡ ∑_{i≤n−1} \frac{t_{1}^{n−1}}{Π · L_{n}^{n−1}} · (aL_{i} + bL_{n} + L_{n} − L_{i}) · ∏_{n≠k≠i}(t_{2} + L_{k})

= ∑_{i≤n−1} \frac{t_{1}^{n−1}}{Π · L_{n}^{n−1}} ((b + 1)L_{n} + (a − 1)L_{i})\frac{t_{2}^{n−1} + σ_{1}^{(n−1)}t_{2}^{n−2} + ⋯}{t_{2} + L_{i}}

= ∑_{i≤n−1} \frac{t_{1}^{n−1}}{Π · L_{n}^{n−1}} ((b + 1)L_{n} + (a − 1)L_{i})·

·(\frac{t_{2}^{n−1} − (−L_{i})^{n−1}}{t_{2} + L_{i}} + σ_{1}^{(n−1)} \frac{t_{2}^{n−2} − (−L_{i})^{n−2}}{t_{2} + L_{i}} + ⋯)+

+ ∑_{i≤n−1} \frac{t_{1}^{n−1}}{Π · L_{n}^{n−1}} ((b + 1)L_{n} + (a − 1)L_{i})\frac{(−L_{i})^{n−1} + σ_{1}^{(n−1)}(−L_{i})^{n−2} + ⋯}{t_{2} + L_{i}}

¿From the first sum we can again eliminate terms from I_{sym}, while the second sum is a rational function in L_{n}, where the numerator and denominator both have degree n, hence this expression is ≡ 0. In particular,

III ≡ − ∑_{i≤n−1} \frac{t_{1}^{n−1}}{Π · L_{n}^{n−1}} ((b + 1)L_{n} + (a − 1)L_{i}) · \frac{t_{2}^{n−1} − (−L_{i})^{n−1}}{t_{2} + L_{i}} .

Therefore, deg(D_{λ}) equals

\frac{n}{a + b} [(at − t + b)^{n−1}(at + b − 1)^{n−1} − (bt + a − 1)^{n−1}(bt + t + a − 1) · \frac{(bt − t + a)^{n−1} − (−1)^{n−1}}{bt − t + a + 1} ]_{t^{n}} .

For n = 3 we recover the polynomial F_{GL(3)} (cf. 5.8) – computed by 7.2 as well.

Also, for arbitrary n but for b = 1 the formula becomes simpler:

deg(D_{λ}) = \frac{n}{(a + 1)^{2}} ((n − 1)a^{n+1} − (n + 1)a^{n−1} + 2(−1)^{n−1}).

This expression coincides with Tevelev’s formula from [?], see also [?, 7.14]. (Notice that in [?, 7.2C], Tevelev provides a formula for arbitrary a and b, which is rather different from ours.)

Example 7.4. Specializations of λ = aL_{1} + bL_{2}. Assume that λ_{s} is a specialization of λ (i.e. λ_{s} is either aL_{1} – obtained by specialization b = 0, or it is aL_{1} + aL_{2} – by taking b = a). In this case, by Corollary 5.8 one has

deg(D_{λ_{s}}) = \frac{ε(λ)|W_{λ}|}{ε(λ_{s})|W_{λ_{s}}|} · ( deg(D_{λ})_{|specialized}).

In the first case, deg(D_{aL_{1}+bL_{2}})|_{b=0} is (−1)^{n}n(n − 1)(a − 1)^{n−1}, and the correction factor is (−1)^{n}(n − 1), hence we recover Boole’s formula

deg(D_{aL_{1}}) = n(a − 1)^{n−1}.

(Evidently, this can be easily deduced by a direct limit computations – as above – as well.)

In the second case, deg(D_{aL_{1}+bL_{2}})|_{b=a} should be divided by −2. We invite the reader to verify that this gives

deg(D_{aL_{1}+aL_{2}}) = \frac{n}{2a} [(t − 1)(at + a − 1)^{n−1} · \frac{(at − t + a)^{n−1} − (−1)^{n−1}}{at − t + a + 1} ]_{t^{n}} .

For a = 1 we recover the well known formula of Holme [?] for the degree of the dual variety of the Grassmannian Gr_{2}(ℂ^{n}):

deg(D_{L_{1}+L_{2}}) = \frac{n}{2} · \frac{1 − (−1)^{n−1}}{2} .

Example 7.5. The case of λ = L_{1} + L_{2} + L_{3}, n ≥ 4. This case was studied by Lascoux [?]. Using K-theory he gave an algorithm to calculate the degree of the dual of the Grassmannian Gr_{3}(ℂ^{n}) and calculated many examples. In Proposition 7.6 we give a closed formula for the degree.

We write \binom{n}{k} for the set of subsets of \{1, 2, …, n\} with k elements. For any S ∈ \binom{n}{k} set L_{S} := ∑_{i∈S} L_{i}. Since T_{λ} = \{L_{i} − L_{j} : i > 3, j ≤ 3\}, one has

f_{λ}^{(n)} · σ_{1} = ∑_{S∈\binom{n}{3}} L_{S} ∏_{{i∉S \atop j∈S}} \frac{L_{S} + L_{i} − L_{j}}{L_{j} − L_{i}} .

Similarly as in the previous examples, we separate L_{n}, and we write the above expression as the sum I +II of two terms, I corresponds to the subsets S with n ∉ S, while II to the others. It is easy to see that lim_{L_{n}→∞} I/L_{n} = 0. In order to analyze the second main contribution, it is convenient to introduce the following expression (in variables L_{1}, …, L_{n} and a new free variable t):

R^{(n)}(t) :=

∑_{J =\{j_{1},j_{2}\}∈\binom{n}{2}}(t+L_{J})(t−L_{j_{1}})(t−L_{j_{2}}) ∏_{i∉J} \frac{(t + L_{j_{1}} + L_{i})(t + L_{j_{2}} + L_{i})(L_{J} + L_{i})}{(L_{j_{1}} − L_{i})(L_{j_{2}} − L_{i})}.

This is a homogeneous expression in variables (L, t) of degree n + 1, hence it can be written as

R^{(n)}(t) = ∑_{k=0}^{n+1} P_{k}^{(n)}t^{n+1−k},

where P_{k}^{(n)} is a symmetric polynomial in variables L. The point is that

II = \frac{R^{(n−1)}(L_{n})}{∏_{i≤n−1} L_{n} − L_{i}} , hence lim II/L_{n} = P_{0}^{(n−1)}.

Therefore,

f_{λ}^{(n)} = P_{0}^{(n−1)} and deg(D_{λ}) = nP_{0}^{(n−1)}/3.

Next, we concentrate on the leading coefficient P_{0}^{(n)} of R^{(n)}(t). For two polynomials R_{1} and R_{2} of degree n + 1, we write R_{1} ≡ R_{2} if their leading coefficients are the same. Set t_{r} := t + L_{j_{r}} for r = 1, 2; and for each k write (over the field ℂ(L))

t_{r}^{k} = Q_{r,k}(t_{r})(t_{r} + L_{j_{1}})(t_{r} + L_{j_{2}}) + A_{r,k}t_{r} + B_{r,k}

for some polynomial Q_{r,k} of degree k − 2 and constants A_{r,k} and B_{r,k}. Then

∏_{i∉J}(t+L_{j_{r}}+L_{i}) = \frac{t_{r}^{n}}{(t_{r} − L_{j_{1}})(t_{r} − L_{j_{2}})}+∑_{k=1}^{n} σ_{k}Q_{r,n−k}+∑_{k=1}^{n} σ_{k} \frac{A_{r,n−k}t_{r} + B_{r,n−k}}{(t_{r} − L_{j_{1}})(t_{r} − L_{j_{2}})}.

In P_{0}^{(n)} the first sum has no contribution since it is in the ideal generated by the (non-constant) symmetric polynomials, the second sum has no contribution either, since its limit is zero when t → ∞. Therefore, R^{(n)}(t) is ≡ with

∑_{J ∈\binom{n}{2}}(t+L_{J})(t−L_{j_{1}})(t−L_{j_{2}})·\frac{(t + L_{j_{1}})^{n}}{(t + 2L_{j_{1}})(t + L_{J})}·\frac{(t + L_{j_{2}})^{n}}{(t + 2L_{j_{2}})(t + L_{J})}·∏_{i∉J} \frac{L_{J} + L_{i}}{(L_{j_{1}} − L_{i})(L_{j_{2}} − L_{i})},

which equals

∑_{J ∈\binom{n}{2}} \frac{1}{t + L_{J}} ·\frac{(t − L_{j_{1}})(t + L_{j_{1}})^{n}}{t + 2L_{j_{1}}} ·\frac{(t − L_{j_{2}})(t + L_{j_{2}})^{n}}{t + 2L_{j_{2}}} ·∏_{i∉J} \frac{L_{J} + L_{i}}{(L_{j_{1}} − L_{i})(L_{j_{2}} − L_{i})}.

Let us define the—binomial-like—coefficients \left\{{n \atop k}\right\} by the expansion (near t = ∞):

(7.6)  \frac{(t − 1)(t + 1)^{n}}{t + 2} = ∑_{k≤n} \left\{{n \atop k}\right\}t^{k},

Using

\frac{1}{t + 1} = \frac{1}{t} − \frac{1}{t^{2}} + \frac{1}{t^{3}} − ⋯,

it provides the expansions

\frac{(t − L_{j_{r}})(t + L_{j_{r}})^{n}}{t + 2L_{j_{r}}} = ∑_{k≤n} \left\{{n \atop k}\right\}t^{k}L_{j_{r}}^{n−k}, and \frac{1}{t + L_{J}} = \frac{1}{t} − \frac{L_{J}}{t^{2}} + \frac{L_{J}^{2}}{t^{3}} − · · · .

Therefore,

(7.7)  P_{0}^{(n)} = ∑_{{k≤n, l≤n \atop k+l≥n+2}}(−1)^{k+l+n}\left\{{n \atop k}\right\}\left\{{n \atop l}\right\}L_{n−k,n−l}^{(n)},

where

L_{n−k,n−l}^{(n)} := \frac{1}{2}· ∑_{J =\{j_{1},j_{2}\}∈\binom{n}{2}} L_{J}^{k+l−n−2}(L_{j_{1}}^{n−k}L_{j_{2}}^{n−l}+L_{j_{1}}^{n−l}L_{j_{2}}^{n−k})·∏_{i∉J} \frac{L_{J} + L_{i}}{(L_{j_{1}} − L_{i})(L_{j_{2}} − L_{i})}.

(Here we already symmetrized L in order to be able to apply in its computation the machinery of symmetric polynomials. Using this index-notation for L_{n−k,n−l}^{(n)} has the advantage that in this way this expression is independent of n, as we will see later.) Next, we determine the constants \left\{{n \atop k}\right\} and L_{n−k,n−l}^{(n)}. The index-set of the sum of (7.7) says that we only need these constants for k ≥ 2 and l ≥ 2.

The constants \left\{{n \atop k}\right\}. The identity

\frac{(t + 1)^{n}(t − 1)}{t + 2} = (t + 1)^{n} + 3 ∑_{i≥1}(−1)^{i}(t + 1)^{n−i}

provides

(7.8)  \left\{{n \atop k}\right\} = \binom{n}{k} + 3 ∑_{i≥1}(−1)^{i}\binom{n − i}{k} for any 0 ≤ k ≤ n.

Notice that these constants satisfy ‘Pascal’s triangle rule’: \left\{{n \atop k}\right\} = \left\{{n−1 \atop k}\right\} + \left\{{n−1 \atop k−1}\right\}. This law together with the ‘initial values’ \left\{{n \atop 0}\right\} = 1 for n even and = −2 for n odd, and with \left\{{n \atop n}\right\} = 1 determines completely all \left\{{n \atop k}\right\} for 0 ≤ k ≤ n. E.g., the first values are:

5 -2 -1 -2 1 2 1
4 1 -2 0 1 1
3 -2 0 0 1
2 1 -1 1
1 -2 1
0 1
n/k 0 1 2 3 4 5

The constants L_{n−k,n−l}^{(n)}. For a ≥ 0, b ≥ 0, n ≥ 3 and a + b = n − 2 we set

X_{a,b}^{(n)} := ∑_{J =\{j_{1},j_{2}\}∈\binom{n}{2}}(L_{j_{1}}^{a}L_{j_{2}}^{b} + L_{j_{1}}^{b}L_{j_{2}}^{a}) · ∏_{i∉J} \frac{L_{J} + L_{i}}{(L_{j_{1}} − L_{i})(L_{j_{2}} − L_{i})}.

Then X_{a,b}^{(n)} = X_{b,a}^{(n)}. By the above homogeneity argument X_{a,b}^{(n)} is a constant. For a > 0 and b > 0 we separate L_{n} and substitute L_{n} = 0, and we get

(7.9)  X_{a,b}^{(n)} = X_{a−1,b}^{(n−1)} + X_{a,b−1}^{(n−1)}.

Therefore, X_{a,b}^{(n)} can be determined from this ‘Pascal rule’ and the values X_{n−1,0}^{(n)} (for n ≥ 3). Next we compute these numbers. We write X_{n−2,0}^{(n)} as I + II + III, where I corresponds to J ∈ \binom{n−1}{2}, and

II = ∑_{j≤n−1} L_{j}^{n−2} ∏_{j≠i≠n} \frac{L_{n} + L_{j} + L_{i}}{(L_{n} − L_{i})(L_{j} − L_{i})},

III = ∑_{j≤n−1} L_{n}^{n−2} ∏_{j≠i≠n} \frac{L_{n} + L_{j} + L_{i}}{(L_{n} − L_{i})(L_{j} − L_{i})}.

Clearly, X_{n−2,0}^{(n)} = lim_{L_{n}→0}(I + II + III). It is easy to see that lim I = 0 and lim II = 1. Moreover, using (7.2), we get

III = \frac{L_{n}^{n−2}}{∏_{i≤n−1}(L_{n} − L_{i})} · ( − \frac{1}{2}R_{2L_{1}}^{(n−1)}(L_{n}) + \frac{3}{2}L_{n} ∑_{j≤n−1} ∏_{j≠i≠n} \frac{L_{j} + L_{i}}{L_{j} − L_{i}} ),

hence

(7.10)  X_{n−2,0}^{(n)} = 1 − \frac{n − 1}{2} + \frac{3}{4}(1 + (−1)^{n}).

Now, we return back to L_{n−k,n−l}^{(n)}. The binomial formula for (L_{j_{1}}+L_{j_{2}})^{k+l−n−2} and (7.9) gives

2L_{n−k,n−l}^{(n)} = ∑_{i=0}^{k+l−n−2} \binom{k + l − n − 2}{i}X_{l−2−i,n−l+i}^{(n)} = X_{l−2,k−2}^{(k+l−2)}.

Let ⟩⟩⟩_{a,b}^{(n)} be defined (for a, b ≥ 0 and a + b = n − 2) by the Pascal rule and initial values ⟩⟩⟩_{n−2,0}^{(n)} = X_{n−2,0}^{(n)} and ⟩⟩⟩_{0,n−2}^{(n)} = 0. Symmetrically, define ⟨⟨⟨_{a,b}^{(n)} = ⟩⟩⟩_{b,a}^{(n)}, hence ⟩⟩⟩_{a,b}^{(n)} + ⟨⟨⟨_{a,b}^{(n)} = X_{a,b}^{(n)}. It is really surprising that X is another incarnation of the constants \left\{{n \atop k}\right\} (for k ≥ 1). Indeed, comparing the initial values of ⟩⟩⟩_{a,b}^{(n)} and \left\{{n \atop k}\right\} we get that

⟩⟩⟩_{l−2,k−2}^{(k+l−2)} = \left\{{k + l − 5 \atop k − 1}\right\} (l ≥ 2, k ≥ 2).

Therefore, we proved the following fact.

Proposition 7.6. For any n ≥ 3 and 2 ≤ k ≤ n consider the contants defined by (7.8), or by (7.6). Consider the weight λ = L_{1} + L_{2} + L_{3} of GL(3) which provides the Plücker embedding of Gr_{3}(ℂ^{n}). Then the degree of the dual variety of Gr_{3}(ℂ^{n}) is

deg(D_{λ}) = \frac{n}{3} · ∑_{\begin{matrix}k≤n−1, \\ 4≤l≤n−1, \\ k+l≥n+1 \end{matrix}}(−1)^{k+l+n−1}\left\{{n − 1 \atop k}\right\} \left\{{n − 1 \atop l}\right\} \left\{{k + l − 5 \atop k − 1}\right\}.

Department of Analysis, Eotvos University Budapest, Hungary

E-mail address: lfeher@renyi.hu

Renyi Mathematical Institute Budapest, Hungary; and Ohio State University, Columbus OH, USA

E-mail address: nemethi@renyi.hu and nemethi@math.ohio-state.edu

Department of Mathematics, University of North Carolina at Chapel Hill, USA

E-mail address: rimanyi@email.unc.edu