Sign hibi cones and pieri algebras for the general linear groups

SIGN HIBI CONES AND PIERI ALGEBRAS FOR THE GENERAL LINEAR GROUPS WANG YI (B.Sc., ECNU, China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2013 To my parents iv v Acknowledgements First of all, I would like to thank my supervisor Professor Lee Soo Teck for his guidance, encouragement and patience. What I have learnt from him is not only mathematics but also the way of life and the attitude to research. I feel very proud to be his student. I would like to thank Professor Roger Howe and Professor Sangjib Kim for their interest in my work. Conversations with them enrich my knowledge and clear my doubts. I would like to thank Ji Feng and Ma Jia Jun for many helpful discussions and suggestions. I learn a lot from them. I would like to thank Gao Rui, Li Xudong and Hou Likun for the two years happy life when we live together. I would like to thank Sun Xiang who helped me solving problems of LATEX. vii viii Acknowledgements I would like to thank Li Shangru and Yuan Zihong for memorable trips. Lastly, I would like to thank my parents for their encouragement and support. Contents Acknowledgements vii Introduction Preliminaries 2.1 Representations of Linear Algebraic Groups . . . . . . . . . . . . . . 2.2 Rational Representations of GLn 2.3 Generalized Iterated Pieri Rules for GLn . . . . . . . . . . . . . . . . 12 2.4 2.5 . . . . . . . . . . . . . . . . . . . . 2.3.1 Polynomial Iterated Pieri Rule . . . . . . . . . . . . . . . . . . 12 2.3.2 Generalized Iterated Pieri Rule . . . . . . . . . . . . . . . . . 15 Posets and Hibi Cones . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4.1 Posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4.2 Increasing Subsets and Decreasing Subsets . . . . . . . . . . . 22 2.4.3 Hibi Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.4 The Hibi Cone Z≥0n,k,h Γ , . . . . . . . . . . . . . . . . . . . . . . 26 Standard Monomial Theory for Hibi Algebras . . . . . . . . . . . . . 30 2.5.1 Standard Monomial Theory . . . . . . . . . . . . . . . . . . . 30 ix x Contents 2.5.2 Semigroup Algebras . . . . . . . . . . . . . . . . . . . . . . . . 31 2.5.3 Semigroup Algebras on Hibi Cones . . . . . . . . . . . . . . . 31 2.6 (GLn , GLk ) Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.7 An Overview of Pieri Algebras . . . . . . . . . . . . . . . . . . . . . . 33 2.8 Polynomial Iterated Pieri Algebras . . . . . . . . . . . . . . . . . . . 35 2.8.1 The Construction of the Polynomial Iterated Pieri Algebras . 35 2.8.2 Polynomials Associated with Tableaux . . . . . . . . . . . . . 37 2.8.3 Monomial Ordering and Sagbi Basis . . . . . . . . . . . . . . . 39 2.8.4 The Structure of Rn,k,h . . . . . . . . . . . . . . . . . . . . . . 42 2.8.5 Reciprocity Algebras . . . . . . . . . . . . . . . . . . . . . . . 44 2.8.6 General Iterated Pieri Algebras . . . . . . . . . . . . . . . . . 45 Sign Hibi Cone 49 3.1 The Structure of the Sign Hibi Cone . . . . . . . . . . . . . . . . . . 49 3.2 Subsemigroups of Sign Hibi Cone . . . . . . . . . . . . . . . . . . . . 53 3.3 The semigroup Ωn,k,l . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4 Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4.1 Expression of Elements in a Sign Hibi Cone . . . . . . . . . . 67 3.4.2 Further Structure of ΩA,B . . . . . . . . . . . . . . . . . . . . 70 Anti-row Iterated Pieri Algebras 73 4.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2 Generators of An,k,l . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.3 Leading Monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.4 Structure of Anti-row Iterated Pieri Algebras . . . . . . . . . . . . . . 91 4.5 Applications to Howe Duality . . . . . . . . . . . . . . . . . . . . . . 93 4.6 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.3 Leading Monomials 89 Definition 4.3.2. For f ∈ Ωn,k,l , define k (0) xfrr(γr mf = (j−1) f (γi ) yij r=1 (j) )−f (γi ) . (4.14) 1≤i≤n, 1≤j≤l Theorem 4.3.2. Let f ∈ Ωn,k,l and let vf be the element of Bn,k,l defined in Definition 4.2.3. Then LM(vf ) = mf . Proof. Define a map m : Ωn,k,l → An,k,l f → mf . For f1 and f2 ∈ Ωn,k,l , mf1 +f2 k (0) +f2 )(γr x(f rr = r=1 (j−1) (f +f2 )(γi ) yij 1≤i≤n, 1≤j≤l k (0) (0) xfrr1 (γr ) xfrr2 (γr = r=1 (j−1) f (γi ) yij1 (j−1) ) f2 (γi yij (j) (j) ) −f1 (γi ) −f2 (γi ) yij yij 1≤i≤n, 1≤j≤l   k  =   (j) )−(f1 +f2 )(γi ) (j−1) (0) f (γ yij1 i xfrr1 (γr ) r=1 1≤i≤n, 1≤j≤l  k (j) (0)  )−f1 (γi )    xfrr2 (γr r=1 (j−1) f (γi ) yij2 (j) )−f2 (γi )   1≤i≤n, 1≤j≤l  = mf1 mf2 . + Therefore, m is a semigroup homomorphism. Note that for χA(c,I) in Gn,k,l , m(χA(c,I) ) = mA(c,I) − , and for −χB(J) in Gn,k,l m(−χB(J) ) = mB(J) . For an arbitrary f ∈ Ωn,k,l , by Corollary 3.3.4, it can be uniquely expressed as a sum p f= q as χA(cs ,Is ) + s=1 bt (−χB(Jt ) ), t=1 90 Chapter 4. Anti-row Iterated Pieri Algebras where as and bt for ≤ s ≤ p and ≤ t ≤ q are positive integers and χA(c1 ,I1 ) ≺ · · · ≺ χA(cp ,Ip ) ≺ −χB(J1 ) ≺ · · · ≺ −χB(Jq ) forms a chain in Gn,k,l . Let vf be as defined in Definition 4.2.3. By Lemma 2.8.3 and Theorem 4.3.1, LM(vf ) = LM t =m mbB(J t) s maA(c s ,Is ) t=1 q bt (−χB(Jt ) ) as χA(cs ,Is ) + t=1 s=1 t=1 s=1 LM(vA(cs ,Is ) ) s=1 p q LM(vB(Jt ) )bt as = t=1 s=1 p = bt vB(J t) as vA(c s ,Is ) q p q p = m(f ) = mf . This proves the theorem. Example 4.3.1. Refer to Example 3.3.3. Recall that f= 2 −1 1 −1 −2 and f = 2χA(1,{1}) + χA(2,{1,2}) + χA(2,{2}) − χB({1,2}) − χB({2}) . By Definition 4.3.1, mA(1,{1}) = x11 y11 , mA(2,{1,2}) = x11 x22 y21 y12 , mA(2,{2}) = x11 x22 y22 , mB({1,2}) = y31 y22 and mB({2}) = y32 . Then LM(vf ) = m2A(1,{1}) mA(2,{1,2}) mA(2,{2}) mB({1,2}) mB({2}) 2 = x411 x222 y11 y21 y31 y12 y22 y32 . By Definition 4.3.2, 2 mf = x411 x222 y11 y21 y31 y12 y22 y32 . l Corollary 4.3.3. For λ ∈ Λ+ n , a Young diagram D with depth(D) ≤ k and α ∈ Z≥0 , Bλ,D,α is a basis of Aλ,D,α . Then Bn,k,l forms a basis of An,k,l . 4.4 Structure of Anti-row Iterated Pieri Algebras Proof. Since #(Bλ,D,α ) = dim(Aλ,D,α ), we only need to prove that all the vf with f ∈ Ωλ,D,α are linearly independent. By Lemma 2.8.2, it suffices to prove that all the vf have distinct leading monomials. By Theorem 4.3.2, it is equivalent to prove that all the mf are distinct. By Definition 4.3.2, mf1 = mf2 if f1 = f2 . Therefore, Bλ,D,α is a basis of Aλ,D,α . By equation 4.4, Bn,k,l forms a basis of An,k,l . 4.4 Structure of Anti-row Iterated Pieri Algebras We can now state the following theorem on the structure of the anti-row iterated Pieri algebra An,k,l . Theorem 4.4.1. Let n, k, l be positive integers such that k ≤ n. (a). An,k,l has a standard monomial theory on Gn,k,l and Bn,k,l is a standard monomial basis for An,k,l . (b). We have LM(An,k,l ) ∼ = Ωn,k,l , so that the initial algebra of An,k,l C[LM(An,k,l )] ∼ = C[Ωn,k,l ]. (c). There exists a flat one-parameter family of C-algebras with general fibre An,k,l and special fibre C[Ωn,k,l ]. Proof. (a). By Corollary 4.3.3, it suffices to prove that Bn,k,l is the set of all standard monomials on Gn,k,l . For each vf ∈ Bn,k,l , by Definition 4.2.3, p q as vA(c s ,Is ) vf = s=1 bt vB(J , t) t=1 where vA(c1 ,I1 ) ≺ · · · ≺ vA(cp ,Ip ) ≺ vB(J1 ) ≺ · · · ≺ vB(Jq ) 91 92 Chapter 4. Anti-row Iterated Pieri Algebras is a chain in Gn,k,l . Therefore, vf is a standard monomial on Gn,k,l . Conversely, let u be a standard monomial on Gn,k,l . Then r2 r1 b as vA(c s ,Is ) u= t vB(J ) t t=1 s=1 where vA(c1 ,I1 ) ≺ . . . ≺ vA(cr1 ,Ir1 ) ≺ vB(J1 ) ≺ . . . ≺ vB(Jr2 ) is a chain in Gn,k,l and as and bt are positive integers for ≤ s ≤ r1 and ≤ t ≤ r2 . Let r1 r2 bt −χB(Jt ) . as χA(cs ,Is ) + f1 = s=1 (4.15) t=1 Then f1 ∈ Ωn,k,l and equation (4.15) is the unique expression of f given in Corollary 3.3.4. Hence, r1 r2 b a v f1 = s vA(c s ,Is ) s=1 t vB(J ) = u. t t=1 Therefore, Bn,k,l is the set of all standard monomials on Gn,k,l . (b). Let h ∈ An,k,l . Then by part (a), there exist f1 , . . . , fp ∈ Ωn,k,l such that p h= ds vfs s=1 with ds ∈ C for ≤ s ≤ p. Then there exists ≤ s0 ≤ p such that LM(h) = max{LM(vfs ) : ≤ s ≤ p} = LM(vfs0 ) = mfs0 . Therefore, LM(An,k,l ) = LM(Bn,k,l ) = {mf : f ∈ Ωn,k,l }. By Lemma 2.8.3, LM(An,k,l ) is a semigroup. Therefore, the map m : f → mf is a semigroup homomorphism from Ωn,k,l to LM(An,k,l ). It is clear that the map mf → f is its inverse. Thus, Ωn,k,l ∼ = LM(An,k,l ) and C[LM(An,k,l )] ∼ = C[Ωn,k,l ]. (c). The algebra C[LM(An,k,l )] is generated by {mf : f ∈ Gn,k,l }. So it is finitely generated. The theorem now follows from Theorem 2.8.4. 4.5 Applications to Howe Duality 4.5 93 Applications to Howe Duality For each positive integer m, let glm = glm (C) be the general Lie algebra of all m × m complex matrices. In this subsection, we consider the lowest weight modules of glk+l which occur in P(Mn,k+l ). For each of these lowest weight modules, we will show that a subset of Bn,k,l can be identified with a basis for the subspace of this module spanned by all the glk highest weight vectors. Definition 4.5.1 ([GW]). Let V be an n-dimensional vector space over C and let x1 , . . . , xn be coordinates on V relative to a basis. We denote by PD(V ) the algebra of polynomial coefficient differential operators on V . This is the subalgebra of End(P(V )) generated (as an associative algebra) by the operators Dxi = ∂ ∂xi and Mxi = multiplication by xi (i = 1, . . . , n). The algebra PD(V ) is called the Weyl algebra. We now let V = Mn,k+l with k ≤ n. Define a GLn action Ad on End(P(Mn,k+l )) by Ad(g)T = ρ1 (g)T ρ1 (g)−1 for g ∈ GLn , T ∈ End(P(Mn,k+l )), where ρ1 is the restriction of the action ρ of GLn × GLk ×Al on P(Mn,k+l ) defined in equation (4.2) to GLn . Then PD(Mn,k+l ) is invariant under GLn relative to the action Ad. We let PD(Mn,k+l )GLn = {T ∈ PD(Mn,k+l ) : Ad(g)T = T for all g ∈ GLn }. (4.16) Recall that the standard coordinates on Mn,k+l defined in equation (4.5). Let n Mij := n Mxsi Mysj = s=1 n ∆ij := Dxti Dytj = t=1 xsi ysj (4.17) ∂ ∂ ∂xti ∂ytj (4.18) s=1 n t=1 94 Chapter 4. Anti-row Iterated Pieri Algebras for ≤ i ≤ k, ≤ j ≤ l and n (x) Eij := n Mxpi Dxpj = p=1 n (y) Est := xpi ∂ ∂xpj (4.19) yqs ∂ ∂yqt (4.20) p=1 n Myqs Dyqt = q=1 p=1 for ≤ i, j ≤ k, ≤ s, t ≤ l. By the First Fundamental Theorem of Invariant Theory for GLn ([Ho89], [GW]), all these operators are elements of PD(Mn,k+l )GLn . Let g := p− + k + p− , (4.21) where p− = Span {Mij : ≤ i ≤ k, ≤ j ≤ l} p+ = Span {∆ij : ≤ i ≤ k, ≤ j ≤ l} n n (x) (y) k = Span Eij + δij : ≤ i, j ≤ k ⊕ Span Est + δst : ≤ s, t ≤ l . 2 Then g is a Lie subalgebra of PD(Mn,k+l )GLn isomorphic to glk+l and it generates PD(Mn,k+l )GLn as an associative algebra ([Go]). Theorem 4.5.1 ([Ho89],[Go]). There is a multiplicity free decomposition of GLn ×g modules given by P(Mn,k+l ) ∼ = ρE,F ⊗ LE,F n k,l , (4.22) E,F where E and F are two Young diagrams such that depth(E) ≤ k, depth(F ) ≤ l, depth(E) + depth(F ) ≤ n and LE,F k,l is an irreducible lowest weight module of g with its lowest weight uniquely determined by (E, F ). By extracting the Un invariants in P(Mn,k+l ), we obtain P(Mn,k+l )Un ∼ = ρE,F n E,F Un ⊗ LE,F k,l . 4.5 Applications to Howe Duality 95 E,F Hence the g representation LE,F k,l can be identified with the ψn -eigenspace of An in P(Mn,k+l )Un . We now identify glk with the following subalgebra of g : (x) glk = Span Eij + n δij : ≤ i, j ≤ k and let (x) n+ k = Span Eij : ≤ i < j ≤ k . Then Un ×Uk An,k,l = P(Mn,k+l ) Un ρE,F n ∼ = ⊗ LE,F k,l n+ k , E,F where LE,F k,l n+ k = v ∈ LE,F k,l : T v = for all T ∈ n+ k E,F is spanned by all glk highest weight vectors in LE,F k,l . In particular, Lk,l n+ k can be identified with the ψnE,F eigenspace of An in An,k,l . Corollary 4.5.2. For two Young diagrams E and F satisfying depth(E) ≤ k, depth(F ) ≤ l and depth(E) + depth(F ) ≤ n, define B(E,F ) = {vf ∈ Bn,k,l : (f (l) )+ = E, (f (l) )−∗ = F }. Then B(E,F ) forms a basis of LE,F k,l n+ k (4.23) . Proof. This follows from Corollary 4.3.3 and the fact that B(E,F ) is contained in the ψnE,F eigenspace of An . 96 Chapter 4. Anti-row Iterated Pieri Algebras 4.6 Future Research We have a standard algebra isomorphism k P(Mn,k+l ) ∼ =P l Cn1,i k ∼ = Cn2,j ⊕ i=1 l P(Cn1,i ) j=1 P(Cn2,j ) , ⊗ i=1 j=1 where Cn1,i and Cn2,j are copies of Cn for ≤ i ≤ k and ≤ j ≤ l. Define the action of (GLn × GL1 )k+l ∼ = GLk+l n ×Ak × Al on P(Mn,k+l ) by ∗ ∗ ∗ ∗ ⊗ · · · ⊗ τn,1 τn,1 ⊗ · · · ⊗ τn,1 ⊗ τn,1 , k where ∗ τn,1 and ∗ τn,1 l are defined in equation (2.20) and equation (2.21) respectively. We restrict the action to GLn ×Ak × Al where GLn = ∆(GLk+l n ) and denote it by (ρ , P(Mn,k+l )). By (GLn , GL1 ) duality (Theorem 2.6.1),     k P(Mn,k+l ) ∼ =  (α ) i) ρ(α ⊗ ρ i  ⊗  n  i=1 l αi ∈Z≥0    ∗ (β ) j) ρ(β ⊗ ρ j  n βj ∈Z≥0 j=1 k ∼ = l s) ρ(α n ⊗ s=1 α=(α1 , .,αk )∈Zk≥0 k ∗ t) ρ(β n t=1 l (α ) ρ1 i ⊗ i=1 (βj ) ⊗ ρ1 j=1 β=(β1 , .,βl )∈Zl≥0 k ∼ = l s) ρ(α n s=1 α=(α1 , .,αk )∈Zk≥0 ∗ t) ρ(β n ⊗ ⊗ ψkα ⊗ ψlβ . t=1 β=(β1 , .,βl )∈Zl≥0 Note that the ψkα × ψlβ eigenspace of Ak × Al in P(Mn,k+l ) is a realization of the tensor product k l s) ρ(α n s=1 ∗ ρn(βt ) ⊗ t=1 , 4.6 Future Research 97 and recall that by equation (2.10), a description on how this tensor product decomposes is called a ((0, 0), k, l)-Pieri rule. We now define En,k,l := P(Mn,k+l )Un . (4.24) Then En,k,l is a module for An ×Ak ×Al and it encodes information on the ((0, 0), k, l)Pieri rule. In a future project, we shall extend the methods and techniques which we have developed in this thesis to study the structure of the algebra En,k,l . In particular, we shall construct a standard monomial basis for En,k,l . Moreover, by Theorem 4.5.1, En,k,l ∼ = ρE,F n Un ⊗ LE,F k,l . E,F Therefore, the ψnE,F -eigenspace of An in En,k,l is a realization of LE,F k,l . Thus using the basis for En,k,l , we will obtain a basis for LE,F k,l . Bibliography [BH] Winfried Bruns, J¨ urgen Herzog, Cohen-Macaulay Rings, Cambridge University Press, 1998,453 pp. [CHV] Aldo Conca, J¨ urgen Herzog and Giuseppe Valla, SAGBI Bases with Applications to Blow-up Algebras, J. Reine Angew. Math. 474 (1996), 113-138. [CLO] David Conca, John Little and Donal O’Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Second edition, Undergraduate Texts in Mathematics, SpringerVerlag, New York, 1997. [FH] William Fulton and Joe Harris, Representation Theory: A First Course, GTM 129, Springer-Verlag New York Inc., 1991. 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Appl., 19, Springer, New York, 1990, 191–225. 101 SIGN HIBI CONES AND PIERI ALGEBRAS FOR THE GENERAL LINEAR GROUPS WANG YI NATIONAL UNIVERSITY OF SINGAPORE 2013 Sign Hibi Cones and Pieri Algebras for the General Linear Groups Wang Yi 2013 [...]... which are needed in the thesis We also discuss the generalized Pieri rule and summarize several existing types of Pieri algebras In particular, we review in some details the structure of polynomial iterated Pieri algebras In Chapter 3, we study the structure of a sign Hibi cone and its subsemigroups Finally, we apply the results of Chapter 3 to study the structure of the anit-row Pieri algebras in Chapter... subsemigroup of a sign Hibi cone ZΓn,l , and is of the form ΩA,B The semigroup Ωn,k,l has a canonical set Gn,k,l of generators For each element in Gn,k,l , we associate with it an element in the algebra An,k,l and let Gn,k,l be the set of elements of An,k,l obtained in this way We show that the set Bn,k,l of standard monomials on Gn,k,l forms a basis for An,k,l , and An,k,l has a flat deformation to the semigroup... 2.3.1 (The Pieri rule) Let D ∈ Λ++ and α ∈ Z≥0 Then n ρD ⊗ ρ(α) = n n ρE n D E |D|+α=|E| For a proof, see [GW] or [Ho95] By iterating the Pieri rule, we obtain the following (see [HL12] and [HKL]): Theorem 2.3.2 (Polynomial Iterated Pieri rule) Let D ∈ Λ++ and α = (α1 , , αh ) ∈ Zh Then n ≥0 h ρD n ρ(αs ) n ⊗ s=1 KF/D,α ρF , n = F 2.3 Generalized Iterated Pieri Rules for GLn where KF/D,α is the. .. Γ∗ if and only if y x in Γ (b) Let P and Q be two posets (i) The direct sum of P and Q is the poset P + Q on the union P ∪ Q such that x y in P + Q if and only if 1) x, y ∈ P and x y in P , or 2) x, y ∈ Q and x y in Q (ii) The ordinal sum of P and Q is the poset P ⊕ Q on P ∪ Q such that x y in P ⊕ Q if and only if 1) x, y ∈ P and x y in P , 2) x, y ∈ Q and x y in Q, or 3) x ∈ P and y ∈ Q (iii) The direct... : g ∈ G} Then the partial ordering on G induces a partial ordering on S A monomial on S of the form vg1 vg2 · · · vgu is called standard if vg1 ≤ vg2 ≤ ≤ vgu The authors proved that the set of standard monomials on S form a vector space basis for An,k,p,h,l Furthermore, An,k,p,h,l has a flat deformation to the semigroup algebra C[H] on H Similar results for Sp2n and On are obtained in the paper... polynomial iterated Pieri algebra If p = h = 0, it is called an anti-row iterated Pieri algebra (which will be studied in this thesis) There are also analogues of Pieri algebras for On = On (C) and Sp2n = Sp2n (C), which are discussed in [KL] In [HKL], the authors studied the structure of the algebra An,k,p,h,l in the stable range, that is, k+p+h+l ≤ n In this case, the structure of the algebra is controlled... In fact, the semistandard tableau T which corresponds to the sequence D0 D2 ··· D1 Dh is defined as follows: We regard F/D as a union of Ds /Ds−1 for 1 ≤ s ≤ h, and put the number s in all the boxes in Ds /Ds−1 Then T is the resulting skew tableau For example, the sequence of Young diagrams corresponds to the semistandard tableau 1 1 2 2 2 Remark 2.3.1 By Lemma 2.3.3, KF/D,α is equal to the number... that the sum of the depth of D and E is at most n (the depth of a Young diagram D is the number of rows of D) ([Ho95]) We denote the representation corresponding to (D, E) by ρD,E , and write ρD,E as ρD when E is the empty diagram There is a n n n combinatorial description of how a tensor product of the form ρD ⊗ ρE decomposes n n It is called the The Littlewood-Richardson Rule ([LR],[Fu],[HL12]) In the. .. D1 D2 ··· Dh and |Ds−1 | + αs = |Ds | for all 1 ≤ s ≤ h There are two other descriptions of the number KF/D,α The first one is related to semistandard tableaux which we now explain Definition 2.3.2 ([Fu]) (a) If D = (d1 , , dn ) and F = (f1 , , fn ) are Young diagrams, then we say that D is contained in F and write D ⊆ F if ds ≤ fs for all 1 ≤ s ≤ n (b) If D ⊆ F , then by removing the boxes of... of D from F , we obtain the skew diagram F/D (c) If we put a positive integer in each box of the skew diagram F/D, then we obtain a skew tableau T and say that the shape of T is F/D (d) If the entries of the skew tableau T is taken from {1, 2, , m} and αj of them are j for 1 ≤ j ≤ m, then we say T has content α = (α1 , , αm ) (e) A skew tableau T is called semistandard if the entries in each row . SIGN HIBI CONES AND PIERI ALGEBRAS FOR THE GENERAL LINEAR GROUPS WANG YI (B.Sc., ECNU, China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL. iterated Pieri algebras. In Chapter 3, we study the structure of a sign Hibi cone and its subsemigroups. Finally, we apply the results of Chapter 3 to study the structure of the anit-row Pieri algebras. 44 2.8.6 General Iterated Pieri Algebras . . . . . . . . . . . . . . . . . 45 3 Sign Hibi Cone 49 3.1 The Structure of the Sign Hibi Cone . . . . . . . . . . . . . . . . . . 49 3.2 Subsemigroups of Sign