A functional realization of the Gelfand-Tsetlin base

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arXiv:2210.12680v1 [math.RT] 23 Oct 2022

УДК 517.986.68

D. V. Artamonov

A functional realization of the Gelfand-Tsetlin base

In the paper we consider a realization of a ﬁnite dimensional irreducible

representation of the Lie algebra glnin the space of functions on the group

GLn. It is proved that functions corresponding to Gelfand-Tsetlin diagrams

are linear combinations of some new functions of hypergeometric type which

are closely related to A-hypergeometric functions. These new functions are

solution of a system of partial diﬀerential equations which one obtains from the

Gelfand-Kapranov-Zelevinsky by an "antisymmetrization". The coeﬃcients in

the constructed linear combination are hypergeometric constants i.e. they are

values of some hypergeometric functions when instead of all arguments ones

are substituted.

Bibliography: 16 items.

Ключевые слова: The Gelfand-Tsetlin base, hypergeometric functions,

the Gelfand-Kapranov-Zelevinsky system.

§ 1. Inroduction

In the year 1950 Gelfand and Tsetlin published a short paper [1], where they gave

an indexation of base vectors in an irreducible ﬁnite-dimensional representation of

the Lie algebra glnand presented formulas for the action of generators of the algebra

in this base. This paper does not contain a derivation of the presented formulas and

it was not translated into English. Nevertheless the results of the paper became

known in the West and there appeared attempts to reproduce the construction of

the base vectors and to reprove the formulas for the action of the generators. In

year 1963 there appeared a paper by Biedenharn and Baird [2], where it was done.

In the Biedenharn’s and Baird’s paper [2] in the case gl3a very interesting

derivation of Gelfand and Tsetlin’s formulas is given. Consider a realization of

a representation in the space of function on the group GL3. Then a function

corresponding to Gelfand-Tsetlin base vectors can be expressed though the Gauss’

hypergeometric function F2,11. And the formulas for the action of generators turn

out to be consequences of the contiguous relations for this function.

In [3] this approach is used to obtain explicit formulas for the Clebsh-Gordan

coeﬃcients for the algebra gl3. Also this approach is used to obtain an explicit

construction of an inﬁnite-dimensional representation of gl3(see [4]). There exist

generalizations of the results of [2] to the case of quantum algebras (see [5], [6]).

Recently their generalization to the case sp4were obtained [7].

In the 60-s it was not possible to obtain a generalization of these construction to

the case glnsince at this time the theory of multivariate hypergeometric function was

1In the paper [2] a realization of a representation using the creation and annihilation operators

is used, in the case gl3these realizations are essentially equivalent

○D. V. Artamonov, 2022

2D. V. ARTAMONOV

not well-developed. The theory of A-hypergeometric functions did not exist at this

time and the system of equation for these functions i.e. the Gelfand-Kapranov-Zelevinsky

system (the GKZ system for short) was not known. All these objects appeared only

in the 80-s of the XX century. In the present paper using these results we do

generalize the results of Biedenharn and Baird to the case of general n.

The main result of the present paper is a formula for a function on the group

that correspond to a Gelfand-Tsetlin base vector (Theorems 5,6). This result makes

possible to give a new derivation of formulas for the action of generators, to obtain

formulas for the Clebsh-Gordan coeﬃcients and so on.

The passage form the case n= 3 to the case n > 3needs new ideas and methods.

In the case n= 3 a formula for a function fcorresponding to a Gelfand-Tsetlin

diagram is derived using a presentation of fas a result of application of lowering

operators to a highest vector. It is not possible to generalize these considerations

in the case gl3to the case gln,n > 3since the formulas for the lowering operators

∇n,k become very complicated (see [8]). Also in the case n>4there appears a new

diﬃculty because of the fact that argumets of the function f, which are minors of

a matrix, are not independent, they satify the Plucker relations.

A possible way to overcome these diﬃculties is to use ideas form the complex

analysis to ﬁnd an analogue in the case n>4of a function corresponding to a

Gelfand-Tsetlin diagram in the case n= 3. This is done in the present paper. A

function of an element g∈GLNby analogy with the case n= 3, is written as a

function of minors of g. We note that in the case gl3the considered function can be

written as an A-hypergeometric function. By analogy with the case gl3for n > 3

we try to ﬁnd the function of interest as an A-hypergeometric function of minors.

This function is deﬁned as a sum of a series called a Γ-series. A Γ-series is a sum

of monomials divided by factorials of exponents. And the set of the exponents of

monomials in this series is a shifted lattice in the space of all possible exponents.

It turns out that it is possible to relate with a Gelfand-Tsetlin diagram a system

of equations that deﬁnes a shifted lattice in the space of exponents (see Section

4.1). Thus to each Gelfand-Tsetlin diagram there corresponds a Γ-series (which

is actually a ﬁnite sum). It is proved that the constructed functions belong to a

canonical embedding of an irreducible ﬁnite dimensional representation into the

functional representation and form a base in it. But this approach does give a

solution of the posed problem. Even in the case gl4the construed functions do not

correspond to Gelfand-Tsetlin base vectors. Nevertheless the constructed base is

related to the Gelfand-Tsetlin base by a transformation which is upper-triangular

relatively some order on diagrams (see Section 6).

In order to prove that the constructed Γ-series form a base in a representation a

new system of PDE is constructed. This system is called the antysimmetrized GKZ

system (A-GKZ for short, see Section 2). We construct a base in the space of it’s

polynomial solutions. It turns out that there is a bijective correspondence between

the constructed Γ-series and the constructed basic solutions of the A-GKZ system

(Theorem 2).

Then we show that the constructed base solutions also belong to a canonical

embedding of an irreducible ﬁnite dimensional representation of glninto the functional

representation, they form a base in it. This base is related to the Gelfand-Tsetlin

THE A-GKZ SYSTEM AND THE GTS BASIS 3

base using a low-triangular transformation (see Section 7.2). We express a function

corresponding to a Gelfand-Tsetlin diagram using these basic solutions.

Since the Gelfand-Tsetlin base is orthogonal relatively an invariant scalar product,

the passage form the base consisting of basic solutions of A-GKZ to the Gelfand-Tsetlin

base is nothing but the orthogonalization transformation. To write this transformation

explicitly one needs to ﬁnd scalar products between basic solutions (see Section 7.1).

When it is done one ﬁnds a low-triangular change of coordinates that diagonalizes

the bilinear form of the considered scalar product (see Section 7.4). Finally when

one has the diagonalizing change of coordinates one can ﬁnd the corresponding

orthogonal base which is nothing but the Gelfand-Tsetlin base (see Theorems 5,6).

In these Theorems the functions corresponding to the Gelfand-Tsetlin base vectors

are expressed through the basic solutions of the A-GKZ system using the numeric

coeﬃcients, which are written as sums of some series. In Section 7.5 we try to

convenience the reader that the obtained formula for the function corresponding to

the Gelfand-Tsetlin diagrams is good enough. The basic solutions of A-GKZ are

Horn’s hypergeometric function. And in Section 7.5 the coeﬃcients occurring in

Theorems 5,6are discussed. It is shown that they are hypergeometric constants

i.e. values of generalized hypergeometric functions (in the Horn’s sense) when one

substitutes ones instead of all their arguments. And these generalized hypergeometric

functions are expressed (see (7.13)) through the Horn functions associated with the

A-hypergeometric function constructed in Section 4.1.

Remark 1. One can overcome some diﬃculties caused by the Plucker relations

by considering as in [2] a realization using the creation and annihilation operators

(or a realization based Weyl construction). But the usage of depended minors has

some fundamental advantages. For example in this case one has a simple description

of the space of functions forming an irreducible representation [8], and also the

presence of the relations suggests some fundamental steps in construction of the

functions of interest.

§ 2. Preliminary facts

In this Section some basic objects and construction are introduced. The Theorem

for the case gl3is formulated. This result was a starting point for the present paper.

2.1. A realization in the space of functions on a group. In the paper Lie

groups and algebras over the ﬁeld Care considered.

Function on the group GLnform a representation of the group GLn. An element

X∈GLnacts onto a function f(g),g∈GLnby a right shift

(Xf )(g) = f(gX).(2.1)

Passing to an inﬁnitesimal version of this action one obtains an action of the Lie

algebra glnon the space of all functions.

Every irreducible ﬁnite-dimensional representation can be realized as a sub-representation

in the space of functions. Let [m1, ..., mn= 0] be a highest weight, then in the

space of functions there exists a highest vector with such a weight which is written

as follows.

4D. V. ARTAMONOV

Let aj

i,i, j = 1, ..., n be a function of a matrix element on the group GLn. Here

jis a row index and iis a column index. Also put

ai1,...,ik:= det(aj

i)j=1,...,k

i=i1,...,ik.(2.2)

That is one takes a determinant of a sub-matrix in a matrix (aj

i), formed by

rows 1, ..., k and columns i1, ..., ik. An operator Ei,j acts onto this determinant by

changing the column indices

Ei,j ai1,...,ik=a{i1,...,ik}|j7→i,(2.3)

where .|j7→iis substitution of jinstead of i. If the index jdoes not occur in

{i1, ..., ik}, then one obtains zero.

Using (2.3), one can easily see that the vector

v0=am1−m2

(m1−m2)!

am2−m3

1,2

(m2−m3)!... amn−1

1,2,...,n−1

mn−1!(2.4)

is a highest vector for the algebra glnwith the weight [m1, m2, ..., mn−1,0]. Thus

one has a canonical embedding of an irreducible ﬁnite-dimensional representation

into the functional representation.

If one considered a highest weight with a non-zero component mnthen in all

formulas below one must change mn−17→ mn−1−mnand multiply all expressions

by amn

1,2,...,n. To make formulas less cumbersome we put mn= 0.

2.2. The Gelfand-Tsetlin base. Consider a chain of subalgebra gln⊃gln−1⊃

... ⊃gl1. Let us be given an irreducible representation Vµnof the algebra glnwith

the highest weight µn. When one restricts the algebra gln↓gln−1the representation

ceases to be irreducible and it splits into a direct sum of irreducible representations

of gln−1. Every irreducible representation of gln−1can occur in this direct sum with

multiplicity not greater than one [8]. Thus one has

Vµn=X

µn−1

Vµn;µn−1,

where µn−1are possible gln−1-highest weight and Vµn;µn−1is a representation of

gln−1with the highest weight µn−1. The sum is taken over all gln−1-highest weights

occurring in the decomposition of Vµninto irreducible representations.

When one continuous restrictions gln−1↓gln−2and so on one gets a splitting of

the following type

V=X

µn−1X

µn−2

... X

µ1

Vµn;...;µ1.(2.5)

Here Vµn;...;µ1is an irreducible representation of the algebra gl1, thus dimVµn;...;µ1=

1. When one chooses base vectors in all Vµn;...;µ1one obtains a base in Vµn. This

is the Gelfand-Tsetlin base.

The base vectors are encoded by sets of highest weights (µn;...;µ1), appearing in

splitting (2.5). If one writes these highest weights one under another in the following

way

THE A-GKZ SYSTEM AND THE GTS BASIS 5







m1,n m2,n ... mn,n

m1,n−1m2,n−1... mn−1,n−1

...

m1,1





,

then one obtains a diagram that is called the Gelfand-Tsetlin diagram. We denote

it as (mi,j ). For elements of this diagram the betweenness condition holds: if a

element of a row occurs between two elements of an upper row then it lies between

them in the numeric sense. The contra verse is also true: every diagram from

which the betweenness condition holds appears as a Gelfand-Tsetlin diagram in the

splitting (2.5).

2.3. A-hypergeometric functions.

2.3.1. AΓ-series. One can ﬁnd information about a Γ-series in [9].

Let B⊂ZNbe a lattice, let γ∈ZNbe a ﬁxed vector. Deﬁne a hypergeometric

Γ-series in variables z1, ..., zNby the formula

Fγ(z) = X

b∈B

zb+γ

Γ(b+γ+ 1),(2.6)

where z= (z1, ..., zN). In the numerator and in the denominator the multi-index

notations are used

zb+γ:=

i=1

zbi+γi

i,Γ(b+γ+ 1) :=

i=1

Γ(bi+γi+ 1).

For the Γ-series considered in the present paper the vectors of exponents b+γ

have integer coordinated. In this case instead of Γ-functions it is reasonable to use

shorter notations with factorials. Hence in the denominator instead of Γ(b+γ+ 1)

we shall write

(b+γ)! :=

i=1

(bi+γi)!

We shall use an agreement that a factorial of a negative integer equals to inﬁnity.

We need the following properties of Γ-series:

1. A vector γcan be changed to γ+b,b∈B, the series remains unchanged.

2. ∂

∂ziFγ,B (z) = Fγ−ei,B (z), где ei= (0, ..., 1на месте i, ..., 0).

3. Let F2,1(a1, a2, b1;z) = Pn∈Z>0

(a1)n(a2)n

(b1)nzn, where (a)n=Γ(a+n)

Γ(a)is the

Gauss hypergeometric series. Then for γ= (−a1,−a2, b1−1,0) and B=

Z<(−1,−1,1,1) >one has

Fγ(z1, z2, z3, z4) = cz−a1

1z−a2

2zb1−1

3F2,1(a1, a2, b1;z3z4

z1z2

)

c=1

Γ(1 −a1)Γ(1 −a2)Γ(b1).

A sum of a Γ-series (when it converges) is called an A-hypergeometric function.

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arXiv:2210.12680v1[math.RT]23Oct2022УДК517.986.68D.V.ArtamonovAfunctionalrealizationoftheGelfand-TsetlinbaseInthepaperweconsiderarealizationofaﬁnitedimensionalirreduciblerepresentationoftheLiealgebraglninthespaceoffunctionsonthegroupGLn.ItisprovedthatfunctionscorrespondingtoGelfand-Tsetlindiagramsar...

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