Classification of Q -homogeneous skew Schur Q -functions
aa r X i v : . [ m a t h . C O ] S e p CLASSIFICATION OF Q -HOMOGENEOUS SKEW SCHUR Q -FUNCTIONS CHRISTOPHER SCHURE
Abstract.
We classify the Q -homogeneous skew Schur Q -functions, i.e., thoseof the form Q λ/µ = k · Q ν . On the way we develop new tools that are usefulalso in the context of other classification problems for skew Schur Q -functions. Keywords. Q -homogeneous, skew Schur Q -function, Schur Q -function Introduction
Schur functions form an important and well studied basis of the algebra of sym-metric functions. They appear in the study of the representations of the symmetricgroups and the general linear groups. In the subalgebra generated by the oddpower sums, the Schur Q -functions form a basis which is less well studied but sharesimilarities to the basis of Schur functions. Stembridge showed in [11] that Schur Q -functions play an analogous role for the irreducible spin characters of the sym-metric groups as the Schur functions do for the ordinary irreducible characters ofthe symmetric groups. In the decomposition of skew Schur Q -functions into Schur Q -functions, the shifted Littlewood-Richardson coefficients appear. The very samecoefficients appear in the decomposition of reduced Clifford products of spin charac-ters into spin characters. In the aforementioned paper, Stembridge proved a shiftedanalogue of the Littlewood-Richardson rule to calculate these coefficients. We willclassify the skew Schur Q -functions that are some multiple of a Schur Q -functionby using this version of the Littlewood-Richardson rule (a different version is givenby Cho in [4]). Our classification is an extension of Salmasian’s classification ofskew Schur Q -functions that are equal to some Schur Q -function in [9].Related classification results were published in the last years. In [10], Stembridgeclassified the multiplicity-free products of Schur functions. Bessenrodt classified the P -multiplicity-free Schur P -functions (some multiple of the Schur Q -functions) in[2] which is a shifted analogue of Stembridge’s result. A classification of multiplicity-free skew Schur functions can be found in [6] by Gutschwager and independentlyin [12] by Thomas and Yong in the context of Schubert calculus. Bessenrodt andKleshchev classified homogeneous skew Schur functions (that is, some multiple of Mathematics Subject Classification.
Primary: 05E05; Secondary: 05E10. a Schur function or a Schur function) in [3]. Our result is a shifted analogue of thisresult.The structure of this note is as follows. In the second section we state requireddefinitions and known properties as well as a new criterion for amenable tableaux(which are the tableaux that are counted in Stembridge’s version of the shiftedLittlewood-Richardson rule). In the third section we develop the tools required forthe proof of the desired classification which includes the decomposition of specificskew Schur Q -functions into Schur Q -functions as well as some properties of theamenable tableaux with the lexicographically largest content and a bijection onskew diagrams that leaves their corresponding skew Schur Q -function unchanged.In the last section we first examine the case of skew Schur Q -functions indexed bydisconnected shifted diagrams before tackling the case of skew Schur Q -functionsindexed by connected shifted diagrams to obtain our result, Theorem 4.21.2. Preliminaries
Partitions, diagrams and tableaux.
We will use notation compatible to[9] and [11].A partition is a tuple λ = ( λ , λ , . . . , λ n ) where λ j ∈ N for all 1 ≤ j ≤ n and λ i ≥ λ i +1 > ≤ i ≤ n −
1. The length of λ is ℓ ( λ ) := n . A partition λ is called a partition of k if | λ | := λ + λ + . . . + λ ℓ ( λ ) = k where | λ | is called the size of λ . A partition with distinct parts is a partition λ = ( λ , λ , . . . , λ n )where λ i > λ i +1 > ≤ i ≤ n −
1. The set of partitions of k with distinctparts is denoted by DP k . By definition, the empty partition ∅ is the only elementin DP and it has length 0. The set of all partitions with distinct parts isdenoted by DP := S k DP k . For some λ ∈ DP the shifted diagram D λ is definedby D λ := { ( i, j ) | ≤ i ≤ ℓ ( λ ) , i ≤ j ≤ i + λ i − } . Convention . In the following we will omit the adjective shifted. This meanswhenever a diagram is mentioned it is always a shifted diagram.A diagram can be depicted as an arrangement of boxes where the coordinates ofthe boxes are interpreted in matrix notation.
Example 2.1.
Let λ = (6 , , , . Then D λ = . . . . . .. . . . .. .. . LASSIFICATION OF Q -HOMOGENEOUS SKEW SCHUR Q -FUNCTIONS 3 Let λ, µ ∈ DP . If ℓ ( µ ) ≤ ℓ ( λ ) and µ i ≤ λ i for all 1 ≤ i ≤ ℓ ( µ ) then the skewdiagram is defined by D λ/µ := D λ \ D µ . Then the size is | D λ/µ | = | D λ | − | D µ | .Each edgewise connected part of the diagram is called a component .The number of components of a diagram D is denoted by comp ( D ). If comp ( D ) =1 the diagram D is called connected , otherwise it is called disconnected .A corner of a diagram D is a box ( x, y ) ∈ D such that ( x + 1 , y ) , ( x, y + 1) / ∈ D . Convention . In the following, if components are numbered, the numbering is asfollows: the first component is the leftmost component, the second component isthe next component to the right of the first component etc.
Example 2.2.
Let λ = (6 , , , and µ = (4 , . Then the diagram is D λ/µ = . .. × . . × . Its size is | D λ/µ | = 7 . The diagram D λ/µ has two components where the firstcomponent consists of three boxes and the second component consists of four boxes.The corners of D λ/µ are the boxes marked × . We consider the alphabet A = { ′ < < ′ < < . . . } . Remark.
The letters 1 , , , . . . are called unmarked letters and the letters denotedby 1 ′ , ′ , ′ , . . . are called marked letters. For a letter x of the alphabet A we denotethe unmarked version of this letter with | x | . Definition 2.3.
Let λ, µ ∈ DP . A tableau T of shape D λ/µ is a map T : D λ/µ →A such thata) T ( i, j ) ≤ T ( i + 1 , j ), T ( i, j ) ≤ T ( i, j + 1) for all i, j ,b) each column has at most one k ( k = 1 , , , . . . ),c) each row has at most one k ′ ( k ′ = 1 ′ , ′ , ′ , . . . ).Let c ( u ) ( T ) = ( c ( u )1 , c ( u )2 , . . . ) where c ( u ) i denotes the number of all letters equal to i in the tableau T for each i . Analogously, let c ( m ) ( T ) = ( c ( m )1 , c ( m )2 , . . . ) where c ( m ) i denotes the number of all letters equal to i ′ in the tableau T for each i . Then the content is defined by c ( T ) = ( c , c , . . . ) := c ( u ) ( T ) + c ( m ) ( T ). If there is some k such that c k > c j = 0 for all j > k then we omit all these c j from c ( T ). Remark.
We depict a tableau T of shape D λ/µ by filling the box ( x, y ) with theletter T ( x, y ) for all x, y . CHRISTOPHER SCHURE
Example 2.4.
Let λ = (8 , , , , and µ = (5 , , . Then a tableau of shape D λ/µ is T = 1 ′ ′ ′
66 7 . We have c ( T ) = (2 , , , , , , . Schur Q -functions. For λ, µ ∈ DP and a countable set of independent vari-ables x , x , . . . the skew Schur Q -function is defined by Q λ/µ := X T ∈ T ( λ/µ ) x c ( T ) where T ( λ/µ ) denotes the set of all tableaux of shape D λ/µ and x ( c ,c ,...,c ℓ ( c ) ) := x c x c · · · with c k := 0 for k > ℓ ( c ). If D µ * D λ then Q λ/µ := 0. Since D λ/ ∅ = D λ ,we denote Q λ/ ∅ by Q λ . Definition 2.5.
Let D be a diagram such that the y th column has no box butthere are boxes to the right of the y th column and after shifting all boxes that areto the right of the y th column one box to the left we obtain a diagram D α/β forsome α, β ∈ DP . Then we call the y th column empty and the diagram D α/β isobtained by removing the y th column. Similarly, let D be a diagram such that the x th row has no box but there are boxes below the x th row and after shifting allboxes that are below the x th row one box up and then all boxes of the diagram onebox to the left we obtain a diagram D α/β for some α, β ∈ DP . Then we call the x th row empty and the diagram D α/β is obtained by removing the x th row. Definition 2.6.
For λ, µ ∈ DP we call the diagram D λ/µ basic if it satisfies thefollowing properties for all 1 ≤ i ≤ ℓ ( µ ): • D µ ⊆ D λ , • ℓ ( λ ) > ℓ ( µ ), • λ i > µ i , • λ i +1 ≥ µ i − D λ/µ has no empty rows or columns.For a given diagram D let ¯ D be the diagram obtained by removing all emptyrows and columns of the diagram D . Since the restrictions of each box in a diagramis unaffected by removing empty rows and columns, there is a content-preservingbijection between tableaux of a given shape and tableaux of the respective shapeobtained by removing empty rows and columns; thus we have Q D = Q ¯ D . Hence,we may restrict our considerations to partitions λ and µ such that D λ/µ is basic. LASSIFICATION OF Q -HOMOGENEOUS SKEW SCHUR Q -FUNCTIONS 5 For some given skew diagram D let the diagram obtained after removing emptyrows and columns be D λ/µ for some λ, µ ∈ DP . Then Q D is equal to the skewSchur Q -function Q λ/µ .For a tableau T of a diagram D the reading word w := w ( T ) is the wordobtained by reading the rows from left to right beginning with the bottom row andending with the top row. The length ℓ ( w ) is the number of letters of the word w and, thus, the number of boxes in D .Let ( x ( i ) , y ( i )) denote the box of the i th letter of the reading word w ( T ). It isthe box that satisfies the property |{ ( u, v ) ∈ D λ/µ | either we have u > x ( i ) or we have u = x ( i ) and v ≤ y ( i ) }| = i. For a reading word w of length n of a tableau the statistics m i ( j ) are defined asfollows: • m i (0) = 0 for all i . • For 1 ≤ j ≤ n the statistic m i ( j ) is equal to the number of times i occursin the word w n − j +1 . . . w n . • For n + 1 ≤ j ≤ n we set m i ( j ) := m i ( n ) + k ( i ) where k ( i ) is the numberof times i ′ occurs in the word w . . . w j − n .Note that c ( u ) i = m i ( n ) and c ( m ) i = m i (2 n ) − m i ( n ).As Stembridge remarked in [11, before Theorem 8.3], the statistics m i ( j ) forsome given i can be calculated simultaneously by taking the word w ( T ) and scanit first from right to left while counting the letters i and afterwards scan it fromleft to right and adding the number of letters i ′ . After the j th step of scanning andcounting the statistic m i ( j ) is calculated. Definition 2.7.
Let k ∈ N and w be a word of length n consisting of letters fromthe alphabet A . The word w is called k -amenable if it satisfies the followingconditions:a) if m k ( j ) = m k − ( j ) then w n − j / ∈ { k, k ′ } for all 0 ≤ j ≤ n − m k ( j ) = m k − ( j ) then w j − n +1 / ∈ { k − , k ′ } for all n ≤ j ≤ n − j is the smallest number such that w j ∈ { k ′ , k } then w j = k ,d) if j is the smallest number such that w j ∈ { ( k − ′ , k − } then w j = k − w is called amenable if it is k -amenable for all k >
1. A tableau T iscalled ( k -)amenable if w ( T ) is ( k -)amenable. Remark.
Definition 2.7 a) can be regarded as follows: Suppose that while scanninga word from right to left we have m k ( j ) = m k − ( j ) for some j < n . Then the nextletter that is scanned cannot be a k ′ or k . CHRISTOPHER SCHURE
Similarly, 2.7 b) can be regarded as follows: Suppose that while scanning a wordfrom left to right we have m k ( j ) = m k − ( j ) for some n ≤ j < n . Then the nextletter that is scanned cannot be a k − k ′ .Definition 2.7 c) states that for a given amenable tableau T the leftmost box ofthe lowermost row with boxes with entry from { k ′ , k } contains a k .Clearly, Definition 2.7 a) ensures m k − ( n ) ≥ m k ( n ) for any k -amenable word oflength n . If m k − ( n ) = m k ( n ) > k − k ′ is scanned before the first ( k − ′ in the word is scanned. Thus, wehave proven the following lemma. Lemma 2.8. [9, Lemma 3.28]
Let w be a k -amenable word for some k > . Let n := ℓ ( w ) . If m k − ( n ) > then m k − ( n ) > m k ( n ) . We will use the shifted version of the Littlewood-Richardson rule by Stembridge.
Proposition 2.9. [11, Theorem 8.3]
For given λ, µ ∈ DP we have Q λ/µ = X ν ∈ DP f λµν Q ν , where f λµν is the number of amenable tableaux T of shape D λ/µ and content ν . For λ ∈ DP , the corresponding Schur P -function is defined by P λ := 2 − ℓ ( λ ) Q λ .In [11, Chapter 8], Stembridge showed that the numbers f λµν above also appear inthe product of P -functions: P µ P ν = X λ ∈ DP f λµν P λ . Using this, one easily obtains the equation f λµν = f λνµ for all λ, µ, ν ∈ DP . Definition 2.10. A border strip is a connected (skew) diagram B such that foreach box ( x, y ) ∈ B the box ( x − , y − / ∈ B . The box ( x, y ) ∈ B such that( x − , y ) / ∈ B and ( x, y + 1) / ∈ B is called the first box of B . The box ( u, v ) ∈ B such that ( u + 1 , v ) / ∈ B and ( u, v − / ∈ B is called the last box of B .A (possibly disconnected) diagram D where all components are border strips iscalled a broken border strip . Then the first box of the rightmost component iscalled the first box of D , and the last box of the leftmost component is called thelast box of D .A ( p, q ) -hook is a set of boxes { ( u, v + q − , . . . , ( u, v + 1) , ( u, v ) , ( u + 1 , v ) , . . . , ( u + p − , v ) } for some u, v ∈ N . Definition 2.11.
Let T be a skew shifted tableau of shape D λ/µ . Define T ( i ) by T ( i ) := { ( x, y ) ∈ D λ/µ | | T ( x, y ) | = i } . LASSIFICATION OF Q -HOMOGENEOUS SKEW SCHUR Q -FUNCTIONS 7 Definition 2.12.
Let T be a tableau. If the last box of T ( i ) is filled with i we call T ( i ) fitting . Remark.
A restatement of 2.7 (c) (respectively, 2.7 (d)) is that T ( k ) (respectively, T ( k − ) is fitting.The remark before Theorem 13.1 of [7] states that for a given tableau T of shape D λ/µ the absolute values of the diagonals from top left to bottom right grow, thatis | T ( x, y ) | < | T ( x + 1 , y + 1) | for all x, y such that ( x, y ) , ( x + 1 , y + 1) ∈ D λ/µ . Aneasy consequence is that each component of T ( i ) is a border strip. This is mentionedin the part after Corollary 8.6 in [8] as well as the fact that each component of T ( i ) has two possible fillings with entries of { i ′ , i } which differ only by the marking ofthe entry in the last box of this component.We will now prove a criterion for k -amenability of a tableau that avoids the useof the reading word and can be used as definition of k -amenability of tableaux. Thefollowing technical definition is required to be able to state Lemma 2.14. Definition 2.13.
Let λ, µ ∈ DP and let T be a tableau of D λ/µ . Then S ⊠ λ/µ ( x, y ) := { ( u, v ) ∈ D λ/µ | u ≤ x, v ≥ y } , S ⊠ T ( x, y ) ( i ) := S ⊠ λ/µ ( x, y ) ∩ T − ( i ) where T − ( i ) denotes the preimage of i, B ( i ) T := { ( x, y ) ∈ D λ/µ | T ( x, y ) = i ′ and T ( x − , y − = ( i − ′ } , d B ( i ) T := { ( x, y ) ∈ D λ/µ | T ( x, y ) = i ′ and T ( x + 1 , y + 1) = ( i + 1) ′ } and b ( i ) T = |B ( i ) T | for all i . Then let B ( i ) T ( d ) denote the set of the first d boxes of B ( i ) T . Lemma 2.14.
Let λ, µ ∈ DP and n := | D λ/µ | . Let T be a tableau of D λ/µ . Thenthe tableau T is k -amenable if and only if either c ( T ) k − = c ( T ) k = 0 or else itsatisfies the following conditions:(1) c ( T ) ( u ) k − > c ( T ) ( u ) k ;(2) when T ( x, y ) = k then |S ⊠ T ( x, y ) ( k − | ≥ |S ⊠ T ( x, y ) ( k ) | ;(3) for each ( x, y ) ∈ B ( k ) T we have |S ⊠ T ( x, y ) ( k − | > |S ⊠ T ( x, y ) ( k ) | ;(4) if d := b ( k ) T + c ( u ) k − c ( u ) k − + 1 > then there is an injective map φ : B ( k ) T ( d ) → \ B ( k − T such that if ( x, y ) ∈ B ( k ) T ( d ) and ( u, v ) = φ ( x, y ) then for all u < r < x we have T ( r, s ) / ∈ { k − , k ′ } for all s such that ( r, s ) ∈ D λ/µ ;(5) T ( k − is fitting;(6) if c ( T ) k > then T ( k ) is fitting.Proof. First we want to show that tableaux that satisfy these conditions are indeed k -amenable. Clearly, such a tableau is k -amenable if c ( T ) k = c ( T ) k − = 0. Hence,we assume that c ( T ) k + c ( T ) k − ≥ CHRISTOPHER SCHURE
Lemma 2.14 (2) ensures that if w i = k then we have m k − ( n − i ) ≥|S ⊠ T ( x ( i ) , y ( i )) ( k − | > |S ⊠ T ( x ( i ) , y ( i )) ( k ) |− m k ( n − i ) since T ( x ( i ) − , y ( i ) − = k if ( x ( i ) − , y ( i ) − ∈ D λ/µ . Lemma 2.14 (3) ensures that if w i = k ′ and( x ( i ) , y ( i )) ∈ B ( k ) T then m k − ( n − i ) > m k ( n − i ). If w i = k ′ and ( x, y ) :=( x ( i ) , y ( i )) / ∈ B ( k ) T then T ( x − , y −
1) = ( k − ′ . But then T ( x − , y ) ∈ { k ′ , k − } .If T ( x − , y ) = k − m k − ( n − j +1) > m k ( n − j +1) where j is definedby ( x ( j ) , y ( j )) = ( x − , y ). But then m k − ( n − i ) > m k ( n − i ). If T ( x − , y ) = k ′ then either ( x − , y ) ∈ B ( k ) T or T ( x − , y −
1) = ( k − ′ . Then we can repeat thisargument until we find a box ( z, y ) where z < x such that either T ( z, y ) = k − z, y ) ∈ B ( k ) T . Thus, it is impossible to have m k − ( i ) = m k ( i ) and w n − i = k ′ forsome i . Hence, we showed that Definition 2.7 (a) is satisfied.Lemma 2.14 (1) ensures that we always have m k − ( n ) > m k ( n ). Let i be suchthat w i = k ′ , T ( x ( i ) − , y ( i ) −
1) = ( k − ′ and m k − ( n + i − > m k ( n + i − j be such that ( x ( j ) , y ( j )) = ( x ( i ) − , y ( i ) − m k − ( n + i ) ≥ m k ( n + i ) and T ( x, z ) > k ′ for all y < z ≤ λ x + x − x, λ x + x − T ( x − , w ) < ( k − ′ for all µ x − + x − ≤ w < y (the leftmost box of this row is ( x − , µ x − + x − m k − ( n + l ) ≥ m k ( n + l ) for all i ≤ l ≤ j −
1. Then m k − ( n + j ) ≥ m k ( n + j )+1 > m k ( n + j ). Hence,Definition 2.7 (b) has not been violated between w i and w j . By this argument, k -amenability of T depends on the boxes ( x, y ) ∈ B ( k ) T . If w i = k ′ and ( x ( i ) , y ( i ))is one of the last c ( u ) k − − c ( u ) k − B ( k ) T then m k − ( n + i ) > m k ( n + i )since m k − ( n ) = m k ( n ) + c ( u ) k − − c ( u ) k . Let w i = k ′ and let the box ( x ( i ) , y ( i )) ∈B ( k ) T ( b ( k ) T + c ( u ) k − c ( u ) k − +1). By Lemma 2.14 (4), there is some j such that w j = ( k − ′ and φ ( x ( i ) , y ( i )) = ( x ( j ) , y ( j )). We have m k − ( n + i ) − m k ( n + i ) ≥ c ( u ) k − − c ( u ) k − ( c ( u ) k − − c ( u ) k − − − k ′ in the box( x ( i ) , y ( i )). Note that pairs of boxes ( s, t ) and ( s +1 , t +1) such that T ( s, t ) = ( k − ′ and T ( s +1 , t +1) = k ′ do not change the difference m k − ( n + i ) − m k ( n + i ) becausethe letter w i = k ′ cannot be between these entries in the reading word and, hence,both letters of such pairs are scanned before we scan w i = k ′ . Also for every box( v, w ) ∈ B ( k ) T ( b ( k ) T + c ( u ) k − c ( u ) k − + 1) such that v > x ( i ) Lemma 2.14 (4) ensuresthat φ ( v, w ) is not in a row above the x ( i ) th row or in the x ( i ) th row to the rightof ( x ( i ) , y ( i )). Hence, T ( v, w ) = k ′ and T ( φ ( v, w )) = ( k − ′ are scanned before w i = k ′ and these entries do not change the difference m k − ( n + i ) − m k ( n + i ). If x ( j ) ≥ x ( i ) then m k − ( n + i ) − m k ( n + i ) > w j = ( k − ′ is scannedbefore w i = k ′ . If x ( j ) < x ( i ) and m k − ( n + i ) − m k ( n + i ) = 0 then w l / ∈ { k − , k ′ } for all i < l < j . Thus, there is no i such that m k − ( n + i −
1) = m k ( n + i −
1) and w i ∈ { k − , k ′ } . Hence, we showed that Definition 2.7 (b) is satisfied. LASSIFICATION OF Q -HOMOGENEOUS SKEW SCHUR Q -FUNCTIONS 9 Lemma 2.14 (5) and Lemma 2.14 (6) are restatements of Definition 2.7 (c) andDefinition 2.7 (d), respectively (as mentioned in the remark after Definition 2.12).In total these conditions ensure k -amenability.Now we want to show that if one of these conditions is not satisfied then T isnot k -amenable. We may assume that a + b > m k − ( n ) ≤ m k ( n ) whichcontradicts Lemma 2.8.Suppose Lemma 2.14 (2) is not satisfied. Let i be such that w i = k is the firstscanned entry k such that ( x, y ) := ( x ( i ) , y ( i )) violates Lemma 2.14 (2). Then T ( x − , y ) = k − |S ⊠ T ( x, y ) ( k − | = |S ⊠ T ( x, y ) ( k ) | −
1. We have to distinguishthe cases T ( x − , y − = k − T ( x − , y −
1) = k −
1. If T ( x − , y − = k − m k − ( n − i ) = |S ⊠ T ( x, y ) ( k − | = |S ⊠ T ( x, y ) ( k ) | − m k ( n − i ) and w i = k which violates Definition 2.7 (a). If T ( x − , y −
1) = k − T ( x − , y ) = k ′ and, therefore, T ( x, y + 1) = k . Then for j such that ( x ( j ) , y ( j )) = ( x − , y ) wemust have m k − ( n − j ) = m k ( n − j ). But then we have m k − ( n − j ) = m k ( n − j )and w j = k ′ which also violates Definition 2.7 (a).Suppose Lemma 2.14 (3) is not satisfied. Let ( x, y ) ∈ B ( k ) T be such that |S ⊠ T ( x, y ) ( k − | ≤ |S ⊠ T ( x, y ) ( k ) | . If T ( x − , y −
1) = k − x, y − ∈ D λ/µ we have that k − T ( x − , y − < T ( x, y − < T ( x, y ) = k ′ which is im-possible. Hence ( x, y − / ∈ D λ/µ and x = y . But then ( x, y ) = ( x, x ) is thelowermost leftmost box of T ( k ) and, since T ( x, x ) = k ′ , this means that T ( k ) is notfitting which violates Definition 2.7 (d). Thus, there is no box ( x, y ) ∈ B ( k ) T suchthat T ( x − , y −
1) = k −
1. Hence, if i is such that ( x, y ) = ( x ( i ) , y ( i )) then m k − ( n − i ) ≤ m k ( n − i ). If m k − ( n − i ) < m k ( n − i ) then T is not k -amenable.If m k − ( n − i ) = m k ( n − i ) then w i = k ′ which violates Definition 2.7 (a).Suppose Lemma 2.14 (4) is not satisfied. Thus, b ( i ) T + c ( u ) k − c ( u ) k − + 1 > x, y ) ∈ B ( k ) T ( b ( i ) T + c ( u ) k − c ( u ) k − + 1) such that each box of B ( k ) T ( b ( i ) T + c ( u ) k − c ( u ) k − + 1) that is below the x th row can be mapped to a differentbox with the given property of Lemma 2.14 (4) but ( x, y ) cannot be mapped inthis way. If i is such that ( x, y ) = ( x ( i ) , y ( i )) then m k − ( n + i ) = m k ( n + i ) since m k − ( n + i ) − m k ( n + i ) = c ( u ) k − − c ( u ) k − ( b ( i ) T − ( b ( i ) T + c ( u ) k − c ( u ) k − + 1)) − s, t ) and ( s + 1 , t + 1) such that T ( s, t ) = ( k − ′ and T ( s + 1 , t + 1) = k ′ do not change the difference m k − ( i ) − m k ( i ) as well as eachbox ( v, w ) ∈ B ( k ) T ( b ( k ) T + c ( u ) k − c ( u ) k − + 1) such that v > x that can be mapped toa different box with the given property of Lemma 2.14 (4) since T ( u, v ) = k ′ and T ( φ ( u, v )) = ( k − ′ are both scanned before the letter w i = k ′ . Since the box( x, y ) cannot be mapped to a box with the given property of Lemma 2.14 (4), thismeans that either there is some l > i such that m k − ( n + l −
1) = m k ( n + l − and w l ∈ { k − , k ′ } , which violates Definition 2.7 (b), or we have m k − ( n − i ) = 0and w i = T ( x ( i ) , y ( i )) = T ( x, y ) = k ′ which violates Definition 2.7 (a).It is clear by definition that a tableau is not k -amenable if Lemma 2.14 (5) andLemma 2.14 (6) are not satisfied.Thus, we showed that the k -amenable tableaux are precisely the ones that satisfythe conditions in Lemma 2.14. (cid:3) Example 2.15.
Let T = × × × × × × × × ′ × × × × × × ′ ′ × × × × × × × × × ′ × ′ ′ be a tableau of shape D (11 , , , , , , / (8 , , , , . We will check the conditions ofLemma 2.14 for k = 2 in the following. We have c ( T ) ( u )1 = 5 > c ( T ) ( u )2 . Since T − (2) = { (2 , , (5 , , (7 , } , we need to check condition (2) of Lemma 2.14for these boxes. We have |S ⊠ T (2 , (1) | = 2 ≥ |S ⊠ T (2 , (2) | , |S ⊠ T (5 , (1) | =3 ≥ |S ⊠ T (5 , (2) | and |S ⊠ T (7 , (1) | = 4 ≥ |S ⊠ T (7 , (2) | . Since B (2) T = { (2 , , (4 , } , we need to check condition (3) of Lemma 2.14 for these boxes. Wehave |S ⊠ T (2 , (1) | = 2 > |S ⊠ T (2 , (2) | and |S ⊠ T (4 , (1) | = 3 > |S ⊠ T (4 , (2) | .Since d := 2+3 − , we have to find a map as in condition (4) of Lemma 2.14for the box (2 , . Such a map is φ ((2 , , . Another one is φ ((2 , , .Clearly, T (1) and T (2) are fitting. Hence, the tableau T is -amenable. It is easy to check that the conditions in the following corollary are included inthe conditions of Lemma 2.14.
Corollary 2.16.
Let λ, µ ∈ DP . Let T be a tableau of shape D λ/µ such that either c ( T ) k = c ( T ) k − = 0 or else it satisfies the following conditions:(1) there is some box ( x, y ) such that T ( x, y ) = k − and T ( z, y ) = k for all z > x ;(2) if T ( x, y ) = k then there is some z < x such that T ( z, y ) = k − ;(3) if T ( x, y ) = k ′ then T ( x − , y −
1) = ( k − ′ ;(4) T ( k − is fitting;(5) if c ( T ) k > then T ( k ) is fitting.Then the tableau is k -amenable. In the following, we will need a specific tableau T λ/µ for a given diagram D λ/µ that was constructed by Salmasian. Definition 2.17. [9, before Lemma 3.5] Let D λ/µ be a skew diagram. The tableau T λ/µ is determined by the following algorithm: LASSIFICATION OF Q -HOMOGENEOUS SKEW SCHUR Q -FUNCTIONS 11 (1) Set k = 1 and U ( λ/µ ) = D λ/µ .(2) Set P k = { ( x, y ) ∈ U k ( λ/µ ) | ( x − , y − / ∈ U k ( λ/µ ) } .(3) For each ( x, y ) ∈ P k set T λ/µ ( x, y ) = k ′ if ( x + 1 , y ) ∈ P k , otherwise set T λ/µ ( x, y ) = k .(4) Let U k +1 ( λ/µ ) = U k ( λ/µ ) \ P k .(5) Increase k by one, and go to (2).We also define P i ( λ/µ ) := P i , for i ≥ Example 2.18.
For λ = (6 , , , and µ = (4 , we have T λ/µ = 1 ′ ′ ′
22 3 . Salmasian proved the amenability of T λ/µ in [9, Lemma 3.9]. Also Corollary 2.16proves amenability of T λ/µ for all λ, µ ∈ DP since T λ/µ is a prototype of a tableausatisfying the conditions of that corollary.3. Properties of the decomposition of skew Schur Q -functions Before we can start to answer the question which skew Schur Q -functions are Q -homogeneous we need to prove some results that will be important in the nextsection.We want to decompose Q λ/ ( n ) for any 1 ≤ n ≤ λ , in particular for the case n = λ −
1, which will be applied in Theorem 4.10.
Definition 3.1.
Let λ ∈ DP . Then the border is defined by B λ := { ( x, y ) ∈ D λ | ( x + 1 , y + 1) / ∈ D λ } . Define B ( n ) λ := { D λ/µ | D λ/µ ⊆ B λ and | D λ/µ | = n } . Definition 3.2.
Let λ ∈ DP . Define E λ to be the set of all partitions whosediagram is obtained after removing a corner in D λ .Using the set B ( n ) λ we can describe the decomposition of Q λ/ ( n ) in general; theset E λ is required for a simpler version of the special case n = 1. We will describeand prove the decomposition of Q λ/ ( n ) and state the special cases n = λ − n = 1 explicitly. Proposition 3.3.
Let λ ∈ DP and ≤ n ≤ λ be an integer. Then Q λ/ ( n ) = X D λ/ν ∈ B ( n ) λ ( D ν ⊆ D λ ) comp ( D λ/ν ) − Q ν . In particular, Q λ/ ( λ − = X ( x,y ) ∈ B × λ c ( x,y ) B λ Q D µ ∪{ ( x,y ) } where D µ = D λ \ B λ , B × λ := { ( x, y ) ∈ B λ | ( x − , y ) / ∈ B λ and ( x, y − / ∈ B λ } and c ( x,y ) B λ = if ( x, y ) is the first or last box of B λ otherwise , and Q λ/ (1) = X ν ∈ E λ Q ν . Proof.
Since f λ ( n ) ν = f λν ( n ) , we need to look at tableaux of shape D λ/ν and content( n ). These n entries from { ′ , } must be in the boxes of B λ . Hence, D λ/ν ∈ B ( n ) λ . Thus, the constituents of Q λ/ ( n ) with a non-zero coefficient are Q ν such that D λ/ν ∈ B ( n ) λ .Each component of D can have two fillings that differ by the marking of the entryof the last box. By definition of amenability, the last box of D λ/ν must contain a 1.Thus, for each component of D λ/ν except for the first one there are two possibilitieshow to fill the last box, giving the stated coefficient. (cid:3) For λ, µ ∈ DP the lexicographical order ≤ in DP is defined as follows: λ ≤ µ whenever either λ = µ or there is some k such that λ i = µ i for 1 ≤ i ≤ k and λ k +1 < µ k +1 where λ k := 0 if k > ℓ ( λ ).The tableau T λ/µ is one of the tableaux of D λ/µ which have the lexicographicallargest content. We are also able to describe how many other tableaux have thesame content as T λ/µ and, hence, we can give the coefficient of the constituentindexed by the lexicographically largest partition, that is c ( T λ/µ ). We want toshow both statements now. In the following, the sets P i = P i ( λ/µ ) are always thesets arising in the construction of T λ/µ . Lemma 3.4.
We have c ( T ) ≤ c ( T λ/µ ) for all amenable tableaux T of shape D λ/µ .However, if c ( T ) = c ( T λ/µ ) then T ( i ) = P i .Proof. In order to obtain the lexicographically largest content of an amenabletableau of shape D λ/µ , we have to insert the maximal number of entries from { ′ , } in D λ/µ , then the maximal number of entries from { ′ , } etc.If | T ( x, y ) | = 1 then ( x − , y − / ∈ D λ/µ . The set of such boxes is P . Thealgorithm of Definition 2.17 fills these boxes only with entries from { ′ , } . Then theentries from { ′ , } must be filled in boxes ( x, y ) such that ( x − , y − / ∈ D λ/µ \ P .The set of such boxes is P and the algorithm of Definition 2.17 fills these boxesonly with entries from { ′ , } . Repeating this argument for all entries greater than2 implies the statement. (cid:3) LASSIFICATION OF Q -HOMOGENEOUS SKEW SCHUR Q -FUNCTIONS 13 Proposition 3.5.
Let D λ/µ be a diagram. Let ν = c ( T λ/µ ) . Then we have f λµν = ℓ ( ν ) Y i =1 comp ( P i ) − . Proof.
Let T be an amenable tableau of D λ/µ with content ν . By Lemma 3.4, wehave T ( i ) = P i . Thus, such a tableau T can differ from T λ/µ only by markings ofsome entries. For each i each component C , . . . , C comp ( P i ) of P i can be filled intwo different ways that differ by the marking of the last box. By Definition 2.7 (c)and (d), the component C must be fitting.Let ( x, y ) be the last box of one of the components C , . . . , C comp ( P i ) and let T ( x, y ) = i ′ . Then, by Corollary 2.16, T is amenable because in this case we have( x − , y − , ( x, y − ∈ P i − and, hence, then T ( x − , y −
1) = ( i − ′ . Thus,for each component of P i except for the first one, there are two possibilities how tofill the last box and the statement follows. (cid:3) Before we can start to classify the Q -homogeneous skew Schur Q -functions weneed to introduce one further operation on diagrams different from the well knowntransposition and rotation. As transposition and rotation of a diagram with specialproperties, this operation keeps the corresponding Schur Q -function unaltered (see[1] for a proof of this fact for transposition and rotation). Definition 3.6.
Let D be a diagram. The orthogonal transpose of D is obtainedas follows: reflect the boxes of D along the diagonal { ( z, − z ) | z ∈ N } . Movethis arrangement of boxes such that the top row with boxes is in the first rowand the lowermost box of the leftmost column with boxes is part of the diagonal { ( z, z ) | z ∈ N } . We denote the orthogonal transpose of a diagram by D ot . Example 3.7.
For D = . . .. . . . .. . . .. . .. we obtain D ot = . .. . . .. . . . .. . .. . . Lemma 3.8.
Let D be a diagram of shape D λ/µ . There is a content-preserving bi-jection between the tableaux of shape D and the tableaux of shape D ot . In particular,we have Q D = Q D ot . Proof.
Let T be a tableau of shape D λ/µ . Let ν := c ( T ) and let n := ℓ ( ν ). Let Λbe the map that maps T to Λ( T ) where Λ( T ) is obtained as follows: • Reflect and move the boxes of T together with their entries along the diag-onal { ( z, − z ) | z ∈ N } . Denote the resulting filling of D otλ/µ by ¯ T . • For all i do the following: – If ¯ T ( x, y ) ∈ { i ′ , i } and ¯ T ( x + 1 , y ) ∈ { i ′ , i } then set Λ( T )( x, y ) =( n − i + 1) ′ . – If ¯ T ( x, y ) ∈ { i ′ , i } and ¯ T ( x, y − ∈ { i ′ , i } then set Λ( T )( x, y ) = n − i + 1. – If ¯ T ( x, y ) ∈ { i ′ , i } and neither ¯ T ( x + 1 , y ) ∈ { i ′ , i } nor ¯ T ( x, y − ∈{ i ′ , i } then if ( x, y ) is the k th such box counted from the left let ( u, v )be the last box of the k th component of T ( i ) . If T ( u, v ) = i ′ setΛ( T )( x, y ) = ( n − i + 1) ′ and if T ( u, v ) = i set Λ( T )( x, y ) = n − i + 1.One can see that Λ maps tableaux of shape D to tableaux of shape D ot . Afterorthogonal transposition, the rows and columns are weakly increasing since weorthogonally transpose the rows and columns and change the entries in reverseorder. Clearly, in Λ( T ) there is at most one i in each column and at most one i ′ ineach row. Hence, the properties of Definition 2.3 are satisfied.Let a be the unmarked version of the least entry from T and b be the unmarkedversion of the greatest entry from T . Then c (Λ( T )) = ¯ ν = ( ν , ν . . . , ν a − , ν b , ν b − , ν b − , . . . , ν a +1 , ν a )where ν = ν = . . . = ν a − = 0.The map Λ is an involution in the set of tableaux and, hence, a bijection.Since Q λ/µ is a symmetric function, there are as many tableaux of shape D λ/µ with content ν as there are with content ¯ ν . Thus, there is a bijection takingtableaux of D λ/µ with content ν to tableaux of D λ/µ with content ¯ ν . Let Θ besuch a bijection. Then Ω := Θ ◦ Λ is a content-preserving bijection since Ω is acomposition of bijections and each of these two bijections flips the content. (cid:3)
Remark.
After proving the previous lemma the author discovered that DeWittproved this result in [5, section 4.2] in a slightly different way (with a minor mistakein the content of the image of a tableau). In her thesis she called this operation“flip” and stated that this operation is well known but unfortunately did not givea reference.For “unshifted” diagrams, that is, diagrams D λ/µ where ℓ ( µ ) = ℓ ( λ ) −
1, or-thogonal transposition is just the concatenation of transposition and rotation. Butunlike transposition and rotation this operation also works for diagrams that arenot unshifted.
LASSIFICATION OF Q -HOMOGENEOUS SKEW SCHUR Q -FUNCTIONS 15 It is possible to give bijections like in Lemma 3.8 for transposition and rotationof diagrams. An example of a bijection for the rotation is to rotate the diagramtogether with the entries through 180 degrees and then change the entries in thesame way as Λ does.4.
Classification of Q -homogeneous skew Schur Q -functions Definition 4.1.
A symmetric function f is called Q -homogeneous if it is somemultiple of a single Schur Q -function, that is, if f = k · Q ν for some ν ∈ DP andsome k ∈ N . A diagram D is called Q -homogeneous if the skew Schur Q -function Q D is Q -homogeneous.We are interested in answering the question which Q λ/µ are Q -homogeneous,that is, for which λ, µ ∈ DP we have Q λ/µ = k · Q ν . Clearly, then we must have ν = c ( T λ/µ ). Proposition 3.5 gives a restriction on the number k in Q λ/µ = k · Q ν ,namely it has to be some power of 2.In the following, we set again P i = P i ( λ/µ ) for i ≥
1; hence P i = T ( i ) λ/µ . Notethat we will always assume that D λ/µ is basic.4.1. The disconnected case.
In the following we will find a classification of the Q -homogeneous skew Schur Q -functions indexed by a disconnected diagram. Wewill first exclude all non- Q -homogeneous skew Schur Q -function indexed by a dis-connected diagram, and then in Proposition 4.10 we will prove the Q -homogeneityof the skew Schur Q -functions indexed by one of the remaining disconnected dia-grams. Lemma 4.2.
Let comp ( D λ/µ ) > and ν = c ( T λ/µ ) . If there is a component C i such that i > and C i has at least two boxes then f λµ ¯ ν > where ¯ ν = ( ν − , ν + 1 ,ν , ν , . . . ) . In particular, Q λ/µ is not Q -homogeneous.Proof. We may consider the case that a component which is not the first componenthas boxes in two rows. Otherwise we may consider the orthogonal transpose of thediagram.Let C i where i > x, y ) is the rightmost box of the lowermost row of C i ∩ P then ( x − , y ) ∈ P and ( x + 1 , y + 1) / ∈ D λ/µ . We obtain a new tableau T if we set T ( x, y ) = 2, T ( x − , y ) = 1 and T ( r, s ) = T λ/µ ( r, s ) for every other box ( r, s ) ∈ D λ/µ . ByCorollary 2.16, T is amenable and has content c ( T ) = ( ν − , ν + 1 , ν , ν , . . . ). (cid:3) Example 4.3.
For T λ/µ = 1 ′
11 1 21 ′ ′ we obtain T = 1 11 2 21 ′ ′ . Lemma 4.4.
Let comp ( D λ/µ ) > and ν = c ( T λ/µ ) . Then we have f λµ ¯ ν > where ¯ ν = ( ν − , ν + 1 , ν , ν , . . . ) . In particular, Q λ/µ is not Q -homogeneous.Proof. Let ( x, y ) be the rightmost box of the lowermost row of C ∩ P . We obtaina new tableau T if we set T ( x, y ) = 2 and T ( r, s ) = T λ/µ ( r, s ) for every other box( r, s ) ∈ D λ/µ . By Corollary 2.16, T is m -amenable for m >
2. There is a 2 but no 1in the y th column. However, there is a 1 in the last box of C ∩ P . Hence, by Lemma2.14, amenability follows. It is clear that c ( T ) = ( ν − , ν + 1 , ν , ν , . . . ). (cid:3) Example 4.5.
For T λ/µ = 11 ′
11 21 ′ ′ we obtain T = 11 ′
12 21 ′ ′ . Lemma 4.6.
Let comp ( D λ/µ ) > and ν = c ( T λ/µ ) . Suppose the leftmost columnof C (which is the leftmost column of D λ/µ ) contains at least two boxes. Then f λµ ¯ ν > where ¯ ν = ( ν − , ν , ν , . . . , ν z , ν z +1 + 1 , ν z +2 , . . . ) where z := ℓ ( λ ) − ℓ ( µ ) .In particular, Q λ/µ is not Q -homogeneous.Proof. Let ( x, x ) be the last box of P . We obtain a new tableau T if we set P ′ := P \ { ( x, x ) } and use this instead of P in the algorithm of Definition 2.17.Let P ′ i := T ( i ) . It is clear that ( x, x ) is the last box of P ′ . If ( x + 1 , x + 1) isthe last box of P then ( x + 1 , x + 1) is the last box of P ′ , etc. Thus, the P ′ i s aredistinguished from the P i s by at most one moved or added box. By Corollary 2.16, T is m -amenable for m >
2. There is a 1 with no 2 below in the last box of C ∩ P .Thus, by Corollary 2.16, T is 2-amenable and, hence, amenable.It is clear that c ( T ) = ν − | P ′ | = | P |−
1. The P i s for all 2 ≤ i ≤ z satisfythe property that the last box is part of the main diagonal { ( a, a ) | a ∈ N } . Asmentioned above, they differ from P ′ i s by the fact that the last box is not ( x + i − ,x + i −
1) but instead ( x + i − , x + i − | P ′ i | = ν i . Then ( x + z − , x + z − P ′ z +1 but since ( x + z, x + z ) / ∈ D λ/µ , it follows | P ′ z +1 | = ν z +1 + 1.Hence, the content is c ( T ) = ( ν − , ν , ν , . . . , ν z , ν z +1 + 1 , ν z +2 , . . . ). (cid:3) Example 4.7.
For T λ/µ = 1 ′
11 21 ′ ′ ′
21 2 ′ we obtain T = 1 ′
11 21 ′ ′
22 2 33 . Lemma 4.8.
Let comp ( D λ/µ ) > and ν = c ( T λ/µ ) . If C has boxes above the rowof the uppermost box of the leftmost column then Q λ/µ is not Q -homogeneous.Proof. Since Lemma 4.6 states that diagrams which have more than one box in theleftmost column are not Q -homogeneous, it suffices to consider diagrams such that LASSIFICATION OF Q -HOMOGENEOUS SKEW SCHUR Q -FUNCTIONS 17 the leftmost column of C has only one box. Let ( t, r ) be the rightmost box of P in the lowermost row of P . Note that the last box of P is to the left of the r th column. We obtain a new tableau T if we modify the algorithm of Definition 2.17so that P ′ := P \ { ( t, r ) } is used instead of P in the algorithm.By Corollary 2.16, the tableau T is m -amenable for m >
2. If T ( t, r ) = 2 then,by Corollary 2.16, this tableau is 2-amenable since T ( t − , r ) = 1. If T ( t, r ) = 2 ′ then we have T ( t − , r − = 1 ′ since ( t − , r − / ∈ D λ/µ . However, there is a 1with no 2 below it in the last box of C ∩ P . Thus, by Lemma 2.14, this tableauis 2-amenable and, hence, amenable. Since | P ′ | = | P | −
1, the content satisfies c ( T ) = ν . (cid:3) Example 4.9.
For T λ/µ = 1 ′
11 21 ′
11 1 1 2 ′ we obtain T = 1 ′
11 21 11 1 2 ′
22 2 3 . Proposition 4.10.
Let λ, µ ∈ DP be such that comp ( D λ/µ ) > and such that D λ/µ is basic. Then Q λ/µ = k · Q ν if and only if k = 2 , λ = ( r + 2 , r, r − , . . . , , µ = ( r + 1) and ν = ( r + 1 , r − , r − , . . . , for some r ≥ .Proof. Let Q λ/µ be Q -homogeneous and D λ/µ be a disconnected diagram. Lemma4.2 states that for 1 < i ≤ comp ( D λ/µ ) every component C i can consist of onlyone box, and Lemma 4.4 states that the diagram must consist of precisely twocomponents. Thus, D λ/µ has only two components C , C where C consists ofa single box. Lemma 4.6 implies that the leftmost column of C can have onlyone box, and Lemma 4.8 yields that this box is in the uppermost row of C . Thisimplies that C has shape D α for some α ∈ DP . The same must be true for theorthogonal transpose of the diagram. Thus, α = ( r, r − , . . . ,
1) for some r ≥ λ = ( r + 2 , r, r − , . . . ,
1) and µ = ( r + 1) = ( λ − B × λ = { (1 , r + 1) } and we obtain ν = ( r + 1 , r − , r − , . . . ,
1) and k = f λµν = 2. (cid:3) Remark.
The case r = 1 in Proposition 4.10 also appeared in [5, Theorem IV.3]where Q -homogeneous skew Schur Q -functions with unshifted diagrams are consid-ered. Example 4.11.
For λ = (6 , , , , and µ = (5) the following two tableaux arethe only amenable tableaux of shape D λ/µ : ′ ,
11 1 1 12 2 23 34 . The connected case.
We have finished the disconnected case and we nowconsider Q -homogeneous skew Schur Q -functions indexed by a connected diagram.The following lemmas show the non- Q -homogeneity of Q λ/µ if some P i in T λ/µ has at least two components. This leads to Lemma 4.16 that shows that in thiscase for Q λ/µ = k · Q ν we obtain k = 1; thus D λ/µ is a “strange” diagram in thesense of Salmasian, classified by him in [9]. Hence this provides the classificationof Q -homogeneous skew Schur Q -functions indexed by a connected diagram. Lemma 4.12.
Let D λ/µ be a diagram. Let ν := c ( T λ/µ ) . Let there be some i > such that comp ( P i ) ≥ and let C , . . . , C comp ( P i ) be the components of P i .Let ( x l , y l ) and ( u l , v l ) be the first box and the last box of C l , respectively. If forsome j ∈ { , , . . . , comp ( P i ) − } we have v j +1 ≥ y j + 2 then f λµ ˜ ν > where ˜ ν = ( ν , ν , . . . , ν i − , ν i − − , ν i + 1 , ν i +1 , ν i +2 , . . . ) .Proof. Let ( u, v ) = ( u j +1 , v j +1 ). Then ( u − , v − , ( u, v − ∈ P i − . Let ( s, v − P i − in the ( v − th column. We obtain a new tableau T ifwe set T ( s, v −
1) = i , T ( s − , v −
1) = i − T ( r, t ) = T λ/µ ( r, t ) for every otherbox ( r, t ) ∈ D λ/µ . If ( s, v ) ∈ D λ/µ then T ( s, v ) = T λ/µ ( s, v ) = i ′ and the propertiesin Definition 2.3 are satisfied. By Corollary 2.16, the tableau T is amenable. It isclear that c ( T ) i − = ν i − − c ( T ) i = ν i + 1 and c ( T ) k = ν k for k = i − , i . (cid:3) Example 4.13.
For λ = (9 , , , , and µ = (6 , , , the changes are writtenin boldface: ′ ′ ′ ′ ′ → ′ ′ ′ ′ . Lemma 4.14.
Let D λ/µ be a diagram. Let ν := c ( T λ/µ ) where ν j := 0 for j > ℓ ( ν ) .Let there be some i > such that comp ( P i ) ≥ and let C , . . . , C comp ( P i ) be thecomponents of P i . Let ( x l , y l ) and ( u l , v l ) be the first box and the last box of C l ,respectively. If for some j ∈ { , , . . . , comp ( P i ) − } we have v j +1 = y j + 1 then f λµ ¯ ν > where ¯ ν = ( ν , ν , . . . , ν i − , ν i − − , ν i , ν i +1 + 1 , ν i +2 , ν i +2 , . . . ) .Proof. Let ( x, y ) = ( x j , y j ) and ( u, y + 1) = ( u j +1 , v j +1 ). Then x > u and we have( x − , y ) , ( x − , y ) ∈ P i − . Let ( s, y ) be the lowermost box of P i in the y th columnand let t be such that T λ/µ ( t, y ) = i −
1. We obtain a new tableau T if we set T ( a, y ) = T λ/µ ( a + 1 , y ) for t − ≤ a ≤ s − T ( s, y ) = ( i + 1) ′ if ( s + 1 , y ) ∈ P i +1 or T ( s, y ) = i + 1 if ( s + 1 , y ) / ∈ P i +1 , and T ( e, f ) = T λ/µ ( e, f ) for every otherbox ( e, f ) ∈ D λ/µ . If ( x − , y + 1) ∈ D λ/µ then T λ/µ ( x − , y + 1) = i ′ , otherwise T λ/µ ( x, y + 1) = i and the boxes of C k and C k +1 are in the same component.By Corollary 2.16, T is m -amenable for m = i, i + 1. There is possibly some b such that T ( b, y ) = i ′ and T ( b − , y − = ( i − ′ . However, there is some LASSIFICATION OF Q -HOMOGENEOUS SKEW SCHUR Q -FUNCTIONS 19 c ≥ b such that T ( c, y −
1) = ( i − ′ and T ( c + 1 , y ) = i ′ . Thus, by Lemma 2.14, i -amenability follows. We possibly have T ( s, y ) = ( i + 1) ′ and T ( s − , y − = i ′ .However, we have T ( u, y +1) = i and there is no i +1 in the ( y +1) th column. Hence,by Lemma 2.14, ( i + 1)-amenability follows. It is clear that c ( T ) i − = ν i − − c ( T ) i +1 = ν i +1 + 1 and c ( T ) j = ν j for j = i − , i + 1. (cid:3) Example 4.15.
For λ = (11 , , , , , , and µ = (7 , , , the changes arewritten in boldface: ′ ′ ′ ′ ′ ′ ′ → ′ ′ ′ ′ ′ ′ ′ . If a diagram is connected then this implies that P must be connected. If ina connected diagram there is some P i where i > Q -function is not Q -homogeneous. Thus, for the Q -homogeneous skew Schur Q -functions the diagram P i must be connected for all i .Using Lemma 3.5 we obtain the following lemma. Lemma 4.16.
Let Q λ/µ = k · Q ν for some k . If comp ( D λ/µ ) = 1 then k = 1 .Remark. As mentioned above, Salmasian [9] classified the “strange” diagrams D λ/µ which are the ones satisfying Q λ/µ = Q ν for some ν ∈ DP . Thus, the Q -homogeneous skew Schur Q -functions to connected diagrams are those on the clas-sification list in [9, Theorem 3.2].We have now completed the classification of the Q -homogeneous skew Schur Q -functions. Since we are also interested in the only constituent of the decomposition,before we state our main Theorem 4.21, we need the following definition and lemma. Definition 4.17.
Let λ = ( λ , λ , . . . , λ ℓ ( λ ) ) ∈ DP . Let µ = ( λ i , λ i , . . . , λ i ℓ ( µ ) )for { i , i , . . . , i ℓ ( µ ) } ⊆ { , , . . . , ℓ ( λ ) } . Then λ \ µ is defined as the partition ob-tained by removing the parts of µ from λ . Example 4.18.
For λ = (9 , , , , , and µ = (5 , , we obtain λ \ µ = (9 , , . Lemma 4.19. If λ = ( a, a − , . . . , and µ is arbitrary then Q λ/µ = Q λ \ µ .Proof. The diagram D otλ/µ is equal to D α for some α ∈ DP . Thus, by Lemma 3.8,we have Q λ/µ = Q D otλ/µ = Q α .We obtain α as follows. We will show that for all 1 ≤ k ≤ a the number k iseither a part of α or a part of µ but it is never a part of both partitions. For this proof only, we will not assume that D λ/µ is basic. This means that in this proof itis possible to have λ = µ .The statement clearly holds for λ = (1). Let λ = ( a, a − , . . . ,
1) with a > D λ/µ .Case 1: (1 , a ) ∈ µ .Then µ = a and the a th column of D λ/µ has at most a − α < a .Let U be the diagram obtained by removing the boxes of the first row.Case 2: (1 , a ) / ∈ µ .Then µ < a and the a th column of D λ/µ has precisely a boxes. Thus, α = a .Let U be the diagram obtained by removing the boxes of the a th column.In both cases we have U = D γ/β for γ = ( a − , a − , . . . ,
1) and some β . Byinduction the statement follows. (cid:3) Example 4.20.
For λ = (5 , , , , and µ = (5 , , the diagram is × × × × ×× × × . × × .. .. , where × denotes a box from D µ .We want to calculate the partition α appearing in the equation Q λ/µ = Q D otλ/µ = Q α . Since (1 , ∈ D µ , there cannot be 5 boxes in the first row of D otλ/µ = D α .Thus, there is a part in µ but not in α . After removing the boxes of the first rowwe obtain × × × . × × .. .. . We have (1 , / ∈ D µ and, thus, there is no part in µ but a part in α . Afterremoving the fourth column we obtain × × ×× × . . We have (1 , ∈ D µ and, thus, there is no part in α but in µ . After removingthe boxes of the first row we obtain × × . . We have (1 , ∈ D µ and, thus, there is no part in α but in µ . After removingthe boxes of the first row we obtain . . LASSIFICATION OF Q -HOMOGENEOUS SKEW SCHUR Q -FUNCTIONS 21 We have (1 , / ∈ D µ and, thus, there is no part in µ but a part in α .We obtain α = (4 ,
1) = (5 , , , , \ (5 , , . Theorem 4.21.
Let λ, µ ∈ DP such that D λ/µ is basic. We have Q λ/µ = k · Q ν ifand only if λ , µ , ν and k satisfy one of the following properties:(i) λ arbitrary, µ = ∅ and ν = λ and k = 1 ,(ii) λ = ( r, r − , . . . , and < ℓ ( µ ) < r − for some m and ν = λ \ µ and k = 1 ,(iii) λ = ( p + q + r, p + q + r − , p + q + r − , . . . , p ) , µ = ( q, q − , . . . , , where p, q, r ≥ and ν = ( p + r + q, p + r + q − , p + r + q − , . . . , p + q + 1 , p + q,p + q − , p + q − , . . . , max { p − q, q + 2 − p } ) and k = 1 ,(iv) λ = ( p + q, p + q − , p + q − , . . ., p + 1 , p ) , µ = ( q, q − , . . . , , where p, q ≥ and ν = ( p + q, p + q − , p + q − , . . . , max { p − q, q − p + 2 } ) and k = 1 ,(v) λ = ( r + 2 , r, r − , . . . , , µ = ( r + 1) and ν = ( r + 1 , r − , r − , . . . , forsome r ≥ and k = 2 .Proof. Case (i) is trivial. For the cases (ii), (iii) and (iv) the proof of homogeneityis the main work of [9] and will not be repeated here. In case (ii), by Lemma 4.19,we have Q λ/µ = Q ( λ/µ ) ot = Q α for D α = D λ \ µ . The partition ν for the cases(iii) and (iv) are easy to deduce since each P i is a hook. Case (v) was shown inProposition 4.10. (cid:3) Acknowledgement.
The QF package for Maple made by John Stembridge ( ) was a helpful tool for analysing thedecomposition of skew Schur Q -functions.This paper is based on the research I did for my master’s thesis and my PhDthesis which were supervised by Prof. Christine Bessenrodt. I am very grateful toChristine Bessenrodt that she introduced me to (skew) Schur Q -functions, and Iwould like to thank her for her help in writing this paper as well as for supervisingmy research. References [1] Farzin Barekat and Stephanie van Willigenburg: Composition of transpositions and equalityof ribbon Schur Q-functions. Electron. J. Combin. 16, P -functions. Ann. Comb. 6,119-124 (2002)[3] Christine Bessenrodt and Alexander S. Kleshchev: On Kronecker products of complex rep-resentations of the symmetric and alternating groups. Pacific J. Math. 190, 201-223 (1999)[4] Soojin Cho: A new Littlewood-Richardson rule for Schur P-functions. Trans. Amer. Math.Soc. 365, 939-972 (2013)[5] Elizabeth A. DeWitt: Identities Relating Schur s-Functions and Q-Functions. Ph. D. Thesis,University of Michigan (2012)[6] Christian Gutschwager: On multiplicity-free skew characters and the Schubert calculus.Ann. Comb. 14, 339-353 (2010) [7] Peter N. Hoffman and John F. Humphreys: Projective Representation of the SymmetricGroups. Oxford Mathematical Monographs, Oxford Science Publications, Clarendon Press(1992)[8] Bruce E. Sagan, Richard P. Stanley: Robinson-Schensted Algorithms for Skew Tableaux. J.Combin. Theory Ser. A 55, 161-193 (1990)[9] Hadi Salmasian: Equality of Schur’s Q-functions and Their Skew Analogues. Ann. Comb.12, 325-346 (2008)[10] John R. Stembridge: Multiplicity-free products of Schur functions. Ann. Comb. 5, 113-121(2001)[11] John R. Stembridge: Shifted Tableaux and the Projective Representations of SymmetricGroups. Adv. Math. 74, 87-134 (1989)[12] Hugh Thomas and Alexander Yong: Multiplicity-free Schubert calculus. Canad. Math. Bull.53, 171-186 (2007)[7] Peter N. Hoffman and John F. Humphreys: Projective Representation of the SymmetricGroups. Oxford Mathematical Monographs, Oxford Science Publications, Clarendon Press(1992)[8] Bruce E. Sagan, Richard P. Stanley: Robinson-Schensted Algorithms for Skew Tableaux. J.Combin. Theory Ser. A 55, 161-193 (1990)[9] Hadi Salmasian: Equality of Schur’s Q-functions and Their Skew Analogues. Ann. Comb.12, 325-346 (2008)[10] John R. Stembridge: Multiplicity-free products of Schur functions. Ann. Comb. 5, 113-121(2001)[11] John R. Stembridge: Shifted Tableaux and the Projective Representations of SymmetricGroups. Adv. Math. 74, 87-134 (1989)[12] Hugh Thomas and Alexander Yong: Multiplicity-free Schubert calculus. Canad. Math. Bull.53, 171-186 (2007)