DOI 10.1007/s10801-010-0269-x
Degree bounds for type-A weight rings and Gelfand–Tsetlin semigroups
Benjamin J. Howard·Tyrrell B. McAllister
Received: 6 November 2009 / Accepted: 17 November 2010 / Published online: 4 December 2010
© Springer Science+Business Media, LLC 2010
Abstract A weight ring in type A is the coordinate ring of the GIT quotient of the variety of flags inCnmodulo a twisted action of the maximal torus inSL(n,C). We show that any weight ring in type A is generated by elements of degree strictly less than the Krull dimension, which is at worstO(n2). On the other hand, we show that the associated semigroup of Gelfand–Tsetlin patterns can have an essential generator of degree exponential inn.
Keywords Weight ring·Weight variety·Cohen–Macaulay ring·Toric degeneration·Gelfand–Tsetlin pattern
1 Introduction
Given a pairλ, μof weights forSLn(C)withλdominant, letVλ[μ]denote theμ- isotropic component of the irreducible representationVλwith highest weightλ. The weight ringR(λ, μ)is the graded ring∞
N=0VN λ[N μ]. We define the weight variety W (λ, μ)as
W (λ, μ):=ProjR(λ, μ).
Second author supported by the Netherlands Organization for Scientific Research (NWO) Mathematics Cluster DIAMANT.
B.J. Howard
Mathematics Department, University of Michigan, Ann Arbor, MI 48109, USA e-mail:[email protected]
T.B. McAllister (
)Department of Mathematics, University of Wyoming, Laramie, WY 82071-2000, USA e-mail:[email protected]
Alternatively, the weight ring is the projective coordinate ring of the GIT quotient of the flag variety modulo theμ-twisted action of the maximal torusTinSLn(C).1
Our first theorem (Theorem3.9 below) is that R(λ, μ) is generated in degree strictly less than the Krull dimension ofR(λ, μ), provided that the degree-one piece Vλ[μ]is nonzero. The basic idea behind the proof is to show that the degree-one piece contains a system of parameters and that thea-invariant ofR(λ, μ)is negative.2(The a-invariant is the degree of the Hilbert series, which is a rational function.) The the- orem then follows from the fact thatR(λ, μ)is Cohen–Macaulay.
It is well known (cf. [8,9,14]) thatR(λ, μ)has a flat degeneration to the semi- group algebraR(λ, μ)of the semigroup of Gelfand–Tsetlin patterns associated to semistandard tableaux of shapemλand contentmμform≥0. In particular, the ring R(λ, μ)is the graded ring associated to a filtration ofR(λ, μ)by natural numbers.
Generators forR(λ, μ)can be lifted to generators ofR(λ, μ), so one might hope that R(λ, μ)is relatively simple. Unfortunately, we find pairsλ, μfor whichR(λ, μ)has essential generators of degree exponential inn.
Our second main result (Theorem5.2below) is that, in the case where n=3k is a multiple of 3, the semigroup algebraR(k3,0)has an essential generator of degree approximately(√
2)n. This is in striking contrast to the upper bound of 2n−8 forR(k3,0), which follows from our first theorem (since the Krull dimension of R(k3,0)is 2n−7). This case is particularly interesting because, via the Gelfand–
MacPherson correspondence,R(k3,0)is the moduli space of n-tuples of points in the projective plane. This is a remarkable example of how a semigroup algebra produced by a promising toric degeneration can fail to serve as an effective proxy for the original ring.
Our motivation for studying the semigroup of Gelfand–Tsetlin patterns was to imitate the method of [12], which considered the case ofnpoints on the projective line. Here one takesλto be a multiple of the second fundamental weight2. It was shown in [12] that the associated semigroup of Gelfand–Tsetlin patterns is generated in degree≤2. We had hoped to use the same method in the case ofnpoints in the projective plane, but the second theorem indicates why this is not the right approach.
However, there might still be another toric degeneration, perhaps among those dis- covered by Caldero [2], that yields a bound better than the one in Theorem3.9
2 A description of the weight ringR(λ, μ)
In this section, we give an explicit description ofR(λ, μ). Letn≥2, and letBdenote the Borel subgroup ofSLn(C)consisting of the upper-triangular matrices inSLn(C).
Fix a nontrivial dominant weightλofSLn(C). We representλas a partition, i.e., as
1Weight varieties for arbitrary reductive Lie groups (not just those of type A) were studied by A. Knutson in his Ph.D. thesis [13]. Knutson also studied the symplectic geometry of these spaces.
2Thea-invariant is negative in all types (see Remark3.8below); however, for types other than type A, the degree-one piece sometimes fails to contain a system of parameters. This condition is equivalent to the con- dition that all semistable flags lie in the supports of degree-oneT-invariants; see [11] for a counterexample inG=SO5(C).
a weakly decreasing sequence(λ1, . . . , λn)of nonnegative integers withλ1≥1 and λn=0. Let μbe a weight ofSLn(C)such thatVλ[μ]is nonzero. Thusμ may be expressed as a sequence(μ1, . . . , μn)of nonnegative integers such thatn
i=1μi= n
i=1λi.
By the Borel–Weil construction, the irreducible representation with highest weight λis the finite-dimensional vector space
Vλ=
holomorphicf: SLn(C)→C|f (gb)=eλ(b)f (g) for allg∈SLn(C)andb∈B
, whereeλ(b):=n
i=1bλiii for b=(bij)1≤i,j≤n∈B. The action ofSLn(C)on Vλ is given by(g·f )(h)=f (g−1h)forg, h∈SLn(C). We define
R(λ):=∞
N=0
VN λ.
Multiplication inR(λ)is the usual multiplication of functionsSLn(C)→C.
Remark 2.1 As we will review in the next section, the dominant weightλdetermines a line bundle Lλ→SLn(C)/B such that the space(SLn(C)/B, Lλ)of sections is isomorphic toVλ. The multiplication inR(λ)coincides with multiplication of the corresponding sections. The ringR(λ)is the coordinate ring of the partial flag variety corresponding to the dominant weightλ.
We define Vλ[μ] :=
f ∈Vλ|f (tg)=eμ(t )f (g)for allt∈T,g∈SLn(C) .
We now define aμ-twisted action ofT⊂SLn(C)onR(λ). Forf ∈VN λ=R(λ)Nof degreeN, the action oft∈Tonf is given by
(t·f )(g):=eN μ(t )f t−1g .
Relative to this twisted action, theT-invariant subring ofR(λ)is exactly R(λ, μ):=
∞ N=0
VN λ[N μ].
Remark 2.2 There is a uniqueSLn(C)-linearization ofLλ. This defines a canonical T-linearization ofLλby restriction toT→SLn(C). The above action ofTcoincides with the canonicalT-linearization twisted byμ.
A fundamental fact from the representation theory ofSLn(C)is thatVλhas a basis indexed by semistandard tableaux of shapeλ. Furthermore, the elements in this basis indexed by semistandard tableaux with contentμform a basis ofVλ[μ]. We briefly review the construction.
A Young diagram of shapeλis a left-justified arrangement ofλ1+ · · · +λnboxes withλi boxes in theith row. For example, ifλ=(3,3,2,1,1,0), then the Young
diagram of shapeλis
A semistandard tableaux of shapeλis a filling of each box in a Young diagram of shapeλwith a number from 1 throughnsuch that the rows are weakly increasing and the columns are strictly increasing. For example, ifλ=(3,3,2,1,1,0), then
is a semistandard tableau of shapeλ.
Such a tableauτ determines a basis vectorbτ∈Vλas follows. Write len(I )for the length of a columnIofτ. We identifyIwith the len(I )-tuple of its entries, read from top to bottom. IfI=(i1, . . . , ilen(I )), let detI: SLn(C)→Cbe the function that re- turns the determinant of the len(I )×len(I )submatrix of entries in rowsi1, . . . , ilen(I ) and columns 1,2, . . . ,len(I ). The basis vectorbτ is then defined by
bτ:=
columnsIofτ
detI.
Hence, in the example above,bτ=det1,2,3,5,6det1,4,5det5,6.
We can also describe the T-isotropic subspaceVλ[μ]in terms of semistandard tableaux. The content of a tableauτ isμ=(μ1, . . . , μn)ifμiis the number of boxes inτ that contain the numberifor 1≤i≤n. The subspaceVλ[μ]is the span of the bτ such thatτ has shapeλand contentμ.
3 The first theorem: generators ofR(λ, μ)
We will derive an upper bound on the degree in whichR(λ, μ) is generated. We generally follow the method of [16] (also explained in [4]). The idea is to find a homogeneous system of parameters, together with an upper bound on thea-invariant of the ring; this yields an upper bound on the degree of a generating set.
We begin by referencing a result of [11] and showing why this implies the ex- istence of a system of parameters in degree one. First we must introduce the no- tion of semistability. Recall thatB is the Borel subgroup ofSLn(C), andSLn(C)/B is the flag variety. A functionf ∈Vλ defines a section of the line bundle Lλ:=
SLn(C)×BC→SLn(C)/B, whereSLn(C)×BCdenotes the quotient ofSLn(C)×C by the equivalence relation(gb, eλ(b)z)∼(g, z). The projectionLλ→SLn(C)/B is given by sending the equivalence class of(g, z)togB. We define theμ-twisted
linearization ofTonLλ byt·(g, z):=(t−1g, eμ(t )z). Givenf ∈Vλ, we define a global sectionsf ofLλbysf(gB)=(g, f (g)). The mapf →sf is an isomorphism Vλ∼=(SLn(C)/B, Lλ). Theμ-twisted torus action onLλ determines an action on global sections, which coincides with the μ-twisted action on Vλ that we defined earlier.
A flaggBis semistable if, for some positive integerN, there is a global section sofLN λwiths(gB)=0 that is invariant under theμ-twisted action ofT. That is, gB is semistable if and only if there is anN >0 and an f ∈VN λ[N μ] such that f (g)=0. It was shown in [11] that we may takeN=1. That is,gBis semistable if and only if there exists anf ∈Vλ[μ]such thatf (g)=0. We will use this fact to show that there is a system of parameters withinVλ[μ]forR(λ, μ).
We now recall some basic facts from commutative algebra. Our main references on Cohen–Macaulay rings and modules are [1,15]. Let k be an algebraically closed field. Suppose thatAis aZ≥0-graded finitely-generated k-algebra withA0=k. Let mdenote the graded ideal generated by the positive degree homogeneous elements ofA. Thenmis the unique graded ideal such that all other graded ideals are contained within it. A homogeneous system of parameters forAis a set of homogeneous ele- mentsx1, . . . , xs such thatsis the Krull dimension ofAand the ideal(x1, . . . , xs)is m-primary. By [1, Theorem 1.5.17] we have thatx1, . . . , xs is a homogeneous system of parameters if and only ifAis an integral extension of the subalgebra k[x1, . . . , xs], and that this is the case if and only ifAis a finitely-generated k[x1, . . . , xs]-module.
Let the null coneN be the subvariety of points in SpecR(λ)at which all positive- degree homogeneous elements ofR(λ, μ)vanish. The result of [11] translates into the following:
Proposition 3.1 The elements of Vλ[μ] suffice to cut out the null cone set- theoretically. That is,N is exactly the set of points at which all elements ofVλ[μ] vanish.
Now suppose that I ⊂R(λ) is the ideal of elements vanishing on the null cone.
By the above proposition, I is the radical closure in R(λ) of the ideal generated byVλ[μ] ⊂R(λ). Recall thatR(λ, μ) is the ring of polynomials inR(λ) that are invariant under theμ-twisted action ofT. Since Tis linearly reductive, there is a canonical projectionπ :R(λ)→R(λ, μ), called the Reynolds operator, which is R(λ, μ)-linear. Following Hilbert (cf. [4, Proposition 3.1]), we have the following result.
Proposition 3.2 The invariant ringR(λ, μ)is a finitely generated module over the subalgebra generated byVλ[μ] ⊂R(λ, μ).
Proof LetJ andS be the ideal and subalgebra, respectively, generated byVλ[μ]in R(λ). Then, sinceI =Rad(J ), we have that Im⊂J for some m >0. SinceT is linearly reductive, the invariant ringR(λ, μ)is finitely generated. Thus, there exist homogeneousy1, . . . , yt ∈R(λ, μ)such thaty1, . . . , yt generateR(λ, μ). Suppose that h1, . . . , h span Vλ[μ]. We have that each yim belongs to the ideal J, and so yim=
j=1fjhj for some homogeneousfj∈R(λ). Now we apply the Reynolds’s
operatorπ to obtainymi =
j=1π(fj)hj. Each coefficientπ(fj)is a homogeneous invariant of degree less thanyim. It follows thatR(λ, μ)is generated as anS-module by monomials m=t
i=1yeii, where eachei < m. There are only a finite number of
such monomials, proving the claim.
We now have the following (cf. [4, Proposition 3.2]):
Proposition 3.3 The degree-one pieceVλ[μ]ofR(λ, μ) contains a homogeneous system of parameters.
Proof We already know from the previous proposition thatR(λ, μ)is a finitely gen- erated module over the subalgebraSgenerated by its degree-one pieceVλ[μ]. Lets be the Krull dimension ofR(λ, μ). Note that the Krull dimension of the subringSof R(λ, μ)is also equal tos, sinceR(λ, μ)is a finitely generated module overS.
By Noether normalization [7, Theorem 13.3], there existf1, . . . , fs ∈Vλ[μ]that are algebraically independent and such thatS is finitely generated over the polyno- mial subringC[f1, . . . , fs] ⊂S. (Thefi may be taken to besgeneric linear combi- nations of a basis ofVλ[μ].)
Since furthermore R(λ, μ) is finitely generated over S, we have that R(λ, μ) is finitely generated overC[f1, . . . , fs]. Finally, we may deduce from the Cayley–
Hamilton theorem [7, Theorem 4.3 and Corollary 4.6] that R(λ, μ) is an integral
extension ofC[f1, . . . , fs].
Proposition 3.4 The weight ringR(λ, μ)is Cohen–Macaulay.
Proof The argument is the same as that given in [15, Corollary 14.25] to show that R(λ)is Cohen–Macaulay. We know that R(λ, μ) has a Gröbner degeneration to a semigroup algebra of Gelfand–Tsetlin patterns (see Sect.4). Such semigroup alge- bras are the invariant subrings of polynomial rings by the action of a torus (cf. [5]).
By the theorem of Hochster [10], the subring of torus invariants in a polynomial ring is Cohen–Macaulay. A general principle regarding Gröbner degenerations is that any good property of the special fiber is shared by the general fiber. This is true in partic- ular for the Cohen–Macaulay property [15, Corollary 8.31].
Proposition 3.5 Iff1, . . . , fs is a homogeneous system of parameters forR(λ, μ), thenR(λ, μ)is a freeC[f1, . . . , fs]-module.
Proof Sincef1, . . . , fsare algebraically independent, the ringC[f1, . . . , fs]is regu- lar, and so this proposition follows from [1, Proposition 2.2.11].
For a graded moduleM, letH (M;t ):=∞
d=0dim(Md)tddenote the Hilbert se- ries ofM. It is well known that, ifMis finitely generated, thenH (M;t )is a rational function int. Leta(M)be the degree ofH (M;t )as a rational function. The number a(M)is called thea-invariant ofM.
Fix a homogeneous system of parametersf1, . . . , fs ∈Vλ[μ] for R(λ, μ). Let S=C[f1, . . . , fs]be the subalgebra generated by thefi. For brevity of notation, we
will writeR:=R(λ, μ). Let f denote thes-tuplef1, . . . , fs. By Theorems 13.37(5) and 13.37(6) of [15],Ris a freeS-module, and
H (R/fR;t )=H (R;t )(1−t )s.
But we can easily computeH (R/fR;t ). Suppose that R=Sy1⊕ · · · ⊕Sym. Let k:=maxj(degyj). Now,H (R/fR;t )is the polynomialp(t )=k
i=0hdtd, wherehd is the number ofyjsuch that degyj=d. Therefore, we have proved the following.
Proposition 3.6 The ring R(λ, μ) is generated in degree ≤k =dimR(λ, μ)+ a(R(λ, μ)).
Proposition 3.7 Thea-invariant ofR(λ, μ)is negative.
Proof LetR:=R(λ, μ). The dimension of thedth graded pieceRd ofRis equal to the number of semistandard tableaux of shapedλand contentdμ; this coincides with the number of integer lattice points in thedth dilate of the rational polytope GT(λ, μ) (see Definition4.1below). As a result of the theory of lattice point enumeration for rational polytopes (see, e.g., [17, Chap. 4]), we may conclude that the Hilbert series H (R;t )=∞
d=0f (d)tdis a rational function of negative degree.
Remark 3.8 In fact, in all types, given a pair of weightsλ, μwithλdominant, the dimension of thedμ-weight space in the irreducible representationVdλwith highest weightdλequals the number of integer lattice points in thedth dilate of a certain polytope (for example, the string polytope associated with the reduced word for the longest Weyl element). And so, in all types, thea-invariants of weight rings are neg- ative.
The above propositions imply our first theorem:
Theorem 3.9 The algebraR(λ, μ)is generated in degree strictly less than the Krull dimension ofR(λ, μ).
Proof This follows immediately from Propositions3.6and3.7.
Finally, we point out that the Krull dimension ofR(λ, μ) is one more than the dimension of the GIT quotient of the flag variety byT. This is at most the dimension of the flag variety itself, which isn(n−1)/2. In the case ofn points in projective spacePm−1, whereλis a multiple of themth fundamental weightmfor SLn(C), the Krull dimension ofR(λ, μ)is at mostn(m−1)−(m2−1)+1.
4 The toric degeneration to Gelfand–Tsetlin patterns
A Gelfand–Tsetlin pattern, or GT pattern, is a triangular array x=(xij)1≤i≤j≤nof real numbers satisfying the interlacing inequalitiesxi,j+1≥xij≥xi+1,j+1. We ex-
press x as a triangular array by arranging the entries as follows:
x1n x2n x3n · · · xnn . . . .
x13 x23 x33
x12 x22
x11
Given a semistandard tableauxτ with entries from 1 throughn, letτ (j )be the tableau obtained fromτ by deleting all boxes containing indices strictly larger thanj. Hence,τ (n)=τ. Letλ(j )denote the shape ofτ (j ). One obtains an integral GT pat- tern x(τ )=(x(τ )ij)1≤i≤j≤n by lettingx(τ )ij =λ(j )i. If τ has shapeλ and con- tent μ=(μ1, . . . , μn), then the resulting GT pattern x(τ ) has top rowλ, and, for 1≤j≤n,
j i=1
x(τ )ij=μ1+ · · · +μj.
We denote this assignment byΦ:τ →x(τ ). It is easy to see that it is a bijection from semistandard tableaux of shapeλand contentμto integral GT patterns with top row λand row sums equal to the partial sums ofμ. The GT patterns with a fixed integral top row and fixed integral row sums constitute a rational polytope.
Definition 4.1 The GT polytope GT(λ, μ)is the set of real GT patterns(xij)1≤i≤j≤n
with top rowλand with row sumsj
i=1xij=μ1+ · · · +μj for 1≤j ≤n.
LetS(λ, μ)denote the graded semigroup of integral GT patterns (under addition) that lie in GT(N λ, N μ)for some nonnegative integerN. Gonciulea and Lakshmibai have described a Gröbner degeneration of the ringR(λ)=∞
N=0VN λto a semigroup algebraR(λ)as the special fiber [9]. It was shown in [14] (and also in [15, Corol- lary 14.24]) that this semigroup is isomorphic to the semigroup of integral GT pat- terns with top rowN λfor some nonnegative integerN. This construction also applies to the subringR(λ, μ)by restricting toT-invariants, as we now describe. See [8] for details.
The resulting degenerated ring R(λ, μ) has the same underlying graded vec- tor space as R(λ, μ). The semistandard tableaux of shape N λ and content N μ, N >0, index a basis for R(λ, μ)N. Let bτ ∈R(λ, μ) denote the basis element indexed byτ. The basis elementbτ is the leading term of bτ ∈R(λ, μ) for a cer- tain filtration ofR(λ, μ)(see [9] and [8,14]). The filtration has the special property that, if τ1, τ2 are any two semistandard tableaux and ifbτ1bτ2 =
τcτbτ, where
the sum is over semistandard tableaux, then the term bΦ−1(x(τ1)+x(τ2)) appears on the right-hand side with coefficient equal to 1. Furthermore, all other termscτbτ
have strictly smaller filtration level. Thus, in R(λ, μ), the multiplication rule be- comes
bτ1bτ2=bΦ−1(x(τ1)+x(τ2)).
ThereforeR(λ, μ) is isomorphic toC[S(λ, μ)], the semigroup algebra of GT pat- terns under addition of patterns.
Given an m-tuple of rational numbersq1, . . . , qm, define den(q1, . . . , qm)to be the least positive integerN such thatN qi∈Zfor eachi, 1≤i≤m. We call this the denominator of the m-tuple. Now, if some vertex x of the polytope GT(λ, μ) has denominator N >1, then the integer point Nx is an essential generator of the semigroup C[S(λ, μ)], since x cannot be written as a sum of other integral patterns in S(λ, μ). In the next section we show the existence of such a vertex with large denominator for the case where n is a multiple of 3, λ= n33, and μ=(1,1, . . . ,1).
5 The second theorem: 3kpoints onP2and a nasty GT pattern
Suppose thatn=3k, wherek≥2 is an integer. Letλ=k3be a multiple of the third fundamental weight forSLn(C). Thus, as a partition,λ=(k, k, k,0, . . . ,0)∈R3k. Now letμbe the “democratic” weight dominated byλ. That is, we representμby the composition(1, . . . ,1)∈R3k. With this choice ofλandμ, the projective variety ProjR(λ, μ)is the moduli space of equally weighted 3k-tuples of points in projective spaceP2(see [6] for more details).
We now construct a GT pattern that we claim will be a vertex of GT(λ, μ). Define the sequences{Tj(1)}and{Tj(2)}by the coupled recurrence relations
T0(1)=k, T0(2)=k−1/2,
Tj(1)=Tj(2)−1−1 (j≥1), Tj(2)=1 2
Tj(1)+Tj(1)−1 (j≥1).
(5.1)
Solving this system of recurrence relations yields the closed-form expressions
Tj(1)=k−2 3j+5
9 −1
2 j
−5
9, (5.2)
Tj(2)=k−2 3j− 5
18 −1
2 j
−2
9. (5.3)
LetN=k+ k/2 −2. We will construct a triangular array x by filling in the entries of x in blocks from the upper left to the lower right using the valuesTj(1)and
Tj(2). Begin by filling the entries in the upper left of the triangular array as follows:
x1n x2n x3n
x1,n−1 x2,n−1 x3,n−1
x1,n−2 x2,n−2 x1,n−3
=
k k k
k k k−1
k k−12 k
We then proceed from the upper left to the lower right of the triangular array by filling in blocks of entries as follows. For 1≤j≤N−1, let
x3,n−2j x2,n−2j−1 x3,n−2j−1 x1,n−2j−2 x2,n−2j−2
x1,n−2j−3
=
Tj(1) Tj(1) Tj(1) Tj(2) Tj(1)
Tj(1)
Ifkis even, the final entries at the bottom of the array are filled in as follows:
x34
x23 x33
x12 x22
x11
=
TN(1) TN(1) TN(1) 2−TN(1) TN(1)
1
On the other hand, ifkis odd, then the final entries are filled in as follows:
x35
x24 x34
x13 x23 x33
x12 x22
x11
=
TN(1) TN(1) TN(1)
TN(1) TN(1) 3−2TN(1) TN(1) 2−TN(1)
1
All the remaining entries of the triangular array are assigned the value 0.
Proposition 5.1 The triangular array constructed above is a vertex of GT(λ, μ)with denominator 2N.
Proof To show that x∈GT(λ, μ), we first check that x is a GT-pattern. In this case, the interlacing inequalities to be verified are
Tj−(1)1> Tj(1)>0 Tj−(1)2> Tj−(2)1> Tj−(1)1
⎫⎬
⎭ for 2≤j≤N,
TN(1)−1>2−TN(1)> TN(1) 2−TN(1)>1> TN(1)
⎫⎬
⎭ ifkis even, TN(1)>3−2TN(1)>0
TN(1)>2−TN(1)>3−2TN(1) TN(1)>1>2−TN(1)
⎫⎪
⎪⎪
⎬
⎪⎪
⎪⎭
ifkis odd.
These are all straightforward consequences of the closed-form expressions (5.2) and (5.3) forTj(1)andTj(2), respectively, so x is a GT-pattern. Thus, to show that x∈ GT(λ, μ), we need only establish that the row-sums of x are correct. This amounts to showing that
Tj(2)−1+Tj(1)−1+Tj(1)=3k−2j,
Tj(1)−1+2Tj(1)=3k−2j−1 (5.4) for 2≤j ≤N. These equalities may be shown using induction and the recursive definition (5.1) ofTj(1)andTj(2). It is clear from (5.4) thatTj(1)has denominator 2j when written as a reduced fraction. Hence, x has denominator 2N, as claimed.
It remains only to show that x is a vertex of GT(λ, μ). We prove this by showing that, for any triangular arrayε, if x±ε∈GT(λ, μ), thenε=0. This is most easily
Fig. 1 Tiling of x whenkis even
seen by partitioning the entries of x so that entries that are equal and adjacent are grouped together. Following [3], we call each group of entries in this partition a tile.
See Fig.1for a depiction of the case whenkis even. Each tile is labeled with the value shared by the entries that it contains.
Suppose that x±ε∈GT(λ, μ). Note that, after the addition of±ε, the entries in each tile must still share a value, and the row-sums must be unchanged [3, The- orem 1.5]. We prove inductively that the entries in each tile cannot have changed, proceeding from the upper left to the lower right.
The entries in the tile labeledT0(1)=kcannot have changed because the top row is fixed. For the same reason, the 0 entries in x are also fixed. Proceeding by induction, the entries in the tile labeledTj(1) cannot have changed because there is a row on which this is the only tile besides the tile labeledTj(1)−1and the tile of 0s, which have already been fixed. Hence, the entries in all the tiles labeledTj(1), 0≤j ≤N, are fixed under the addition of±ε. Finally, for 1≤j ≤N−1, the tile labeledTj(2)lies on a row in which the other entries (Tj(1),Tj(1)+1, and 0) have been shown to be fixed, so the entry in this tile is also fixed under the addition of±ε. Therefore, we conclude
thatε=0, so that x is a vertex, as claimed.
The following theorem is an immediate consequence.
Theorem 5.2 The Gelfand–Tsetlin algebraR(k3, μ)has essential generators of degree exceeding 2(n/2)−3.
Acknowledgements We thank Harm Derksen, Ionu¸t Ciocan-Fontanine, and Mircea Musta¸t˘a for their invaluable advice. We also thank the anonymous referees for several helpful suggestions.
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